Daily Papers

Recently, many methods have been developed to extend the context length of pre-trained large language models (LLMs), but they often require fine-tuning at the target length (gg4K) and struggle to effectively utilize information from the middle part of the context. To address these issues, we propose Continuity-Relativity indExing with gAussian Middle (CREAM), which interpolates positional encodings by manipulating position indices. Apart from being simple, CREAM is training-efficient: it only requires fine-tuning at the pre-trained context window (eg, Llama 2-4K) and can extend LLMs to a much longer target context length (eg, 256K). To ensure that the model focuses more on the information in the middle, we introduce a truncated Gaussian to encourage sampling from the middle part of the context during fine-tuning, thus alleviating the ``Lost-in-the-Middle'' problem faced by long-context LLMs. Experimental results show that CREAM successfully extends LLMs to the target length for both Base and Chat versions of Llama2-7B with ``Never Miss A Beat''. Our code will be publicly available soon.

In learned image compression, probabilistic models play an essential role in characterizing the distribution of latent variables. The Gaussian model with mean and scale parameters has been widely used for its simplicity and effectiveness. Probabilistic models with more parameters, such as the Gaussian mixture models, can fit the distribution of latent variables more precisely, but the corresponding complexity will also be higher. To balance between compression performance and complexity, we extend the Gaussian model to the generalized Gaussian model for more flexible latent distribution modeling, introducing only one additional shape parameter, beta, than the Gaussian model. To enhance the performance of the generalized Gaussian model by alleviating the train-test mismatch, we propose improved training methods, including beta-dependent lower bounds for scale parameters and gradient rectification. Our proposed generalized Gaussian model, coupled with the improved training methods, is demonstrated to outperform the Gaussian and Gaussian mixture models on a variety of learned image compression methods.

Recently, 3D Gaussian splatting (3D-GS) has achieved great success in reconstructing and rendering real-world scenes. To transfer the high rendering quality to generation tasks, a series of research works attempt to generate 3D-Gaussian assets from text. However, the generated assets have not achieved the same quality as those in reconstruction tasks. We observe that Gaussians tend to grow without control as the generation process may cause indeterminacy. Aiming at highly enhancing the generation quality, we propose a novel framework named GaussianDreamerPro. The main idea is to bind Gaussians to reasonable geometry, which evolves over the whole generation process. Along different stages of our framework, both the geometry and appearance can be enriched progressively. The final output asset is constructed with 3D Gaussians bound to mesh, which shows significantly enhanced details and quality compared with previous methods. Notably, the generated asset can also be seamlessly integrated into downstream manipulation pipelines, e.g. animation, composition, and simulation etc., greatly promoting its potential in wide applications. Demos are available at https://taoranyi.com/gaussiandreamerpro/.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multi-task learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty on both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performances when dealing with group-structured data. The model handles irregular grid of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real datasets. The overall algorithm, called MagmaClust, is publicly available as an R package.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

We present SplattingAvatar, a hybrid 3D representation of photorealistic human avatars with Gaussian Splatting embedded on a triangle mesh, which renders over 300 FPS on a modern GPU and 30 FPS on a mobile device. We disentangle the motion and appearance of a virtual human with explicit mesh geometry and implicit appearance modeling with Gaussian Splatting. The Gaussians are defined by barycentric coordinates and displacement on a triangle mesh as Phong surfaces. We extend lifted optimization to simultaneously optimize the parameters of the Gaussians while walking on the triangle mesh. SplattingAvatar is a hybrid representation of virtual humans where the mesh represents low-frequency motion and surface deformation, while the Gaussians take over the high-frequency geometry and detailed appearance. Unlike existing deformation methods that rely on an MLP-based linear blend skinning (LBS) field for motion, we control the rotation and translation of the Gaussians directly by mesh, which empowers its compatibility with various animation techniques, e.g., skeletal animation, blend shapes, and mesh editing. Trainable from monocular videos for both full-body and head avatars, SplattingAvatar shows state-of-the-art rendering quality across multiple datasets.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Most advances in 3D Generative Adversarial Networks (3D GANs) largely depend on ray casting-based volume rendering, which incurs demanding rendering costs. One promising alternative is rasterization-based 3D Gaussian Splatting (3D-GS), providing a much faster rendering speed and explicit 3D representation. In this paper, we exploit Gaussian as a 3D representation for 3D GANs by leveraging its efficient and explicit characteristics. However, in an adversarial framework, we observe that a na\"ive generator architecture suffers from training instability and lacks the capability to adjust the scale of Gaussians. This leads to model divergence and visual artifacts due to the absence of proper guidance for initialized positions of Gaussians and densification to manage their scales adaptively. To address these issues, we introduce a generator architecture with a hierarchical multi-scale Gaussian representation that effectively regularizes the position and scale of generated Gaussians. Specifically, we design a hierarchy of Gaussians where finer-level Gaussians are parameterized by their coarser-level counterparts; the position of finer-level Gaussians would be located near their coarser-level counterparts, and the scale would monotonically decrease as the level becomes finer, modeling both coarse and fine details of the 3D scene. Experimental results demonstrate that ours achieves a significantly faster rendering speed (x100) compared to state-of-the-art 3D consistent GANs with comparable 3D generation capability. Project page: https://hse1032.github.io/gsgan.

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

A neural architecture with randomly initialized weights, in the infinite width limit, is equivalent to a Gaussian Random Field whose covariance function is the so-called Neural Network Gaussian Process kernel (NNGP). We prove that a reproducing kernel Hilbert space (RKHS) defined by the NNGP contains only functions that can be approximated by the architecture. To achieve a certain approximation error the required number of neurons in each layer is defined by the RKHS norm of the target function. Moreover, the approximation can be constructed from a supervised dataset by a random multi-layer representation of an input vector, together with training of the last layer's weights. For a 2-layer NN and a domain equal to an n-1-dimensional sphere in {mathbb R}^n, we compare the number of neurons required by Barron's theorem and by the multi-layer features construction. We show that if eigenvalues of the integral operator of the NNGP decay slower than k^{-n-2{3}} where k is an order of an eigenvalue, then our theorem guarantees a more succinct neural network approximation than Barron's theorem. We also make some computational experiments to verify our theoretical findings. Our experiments show that realistic neural networks easily learn target functions even when both theorems do not give any guarantees.

While 3D Gaussian Splatting has recently become popular for neural rendering, current methods rely on carefully engineered cloning and splitting strategies for placing Gaussians, which can lead to poor-quality renderings, and reliance on a good initialization. In this work, we rethink the set of 3D Gaussians as a random sample drawn from an underlying probability distribution describing the physical representation of the scene-in other words, Markov Chain Monte Carlo (MCMC) samples. Under this view, we show that the 3D Gaussian updates can be converted as Stochastic Gradient Langevin Dynamics (SGLD) updates by simply introducing noise. We then rewrite the densification and pruning strategies in 3D Gaussian Splatting as simply a deterministic state transition of MCMC samples, removing these heuristics from the framework. To do so, we revise the 'cloning' of Gaussians into a relocalization scheme that approximately preserves sample probability. To encourage efficient use of Gaussians, we introduce a regularizer that promotes the removal of unused Gaussians. On various standard evaluation scenes, we show that our method provides improved rendering quality, easy control over the number of Gaussians, and robustness to initialization.

We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is Gaussian (or more generally any strongly log-concave distribution) in d dimensions and the noise model is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error opt + epsilon for any strongly log-concave target distribution. For adversarial noise, our tester-learner obtains error O(opt) + epsilon in polynomial time when the target distribution is Gaussian; for strongly log-concave distributions, we obtain O(opt) + epsilon in quasipolynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. (2023). This enables us to simulate a variant of the algorithm of Diakonikolas et al. (2020) for learning noisy halfspaces using nonconvex SGD but in the testable learning setting.

Approximate inference in Gaussian process (GP) models with non-conjugate likelihoods gets entangled with the learning of the model hyperparameters. We improve hyperparameter learning in GP models and focus on the interplay between variational inference (VI) and the learning target. While VI's lower bound to the marginal likelihood is a suitable objective for inferring the approximate posterior, we show that a direct approximation of the marginal likelihood as in Expectation Propagation (EP) is a better learning objective for hyperparameter optimization. We design a hybrid training procedure to bring the best of both worlds: it leverages conjugate-computation VI for inference and uses an EP-like marginal likelihood approximation for hyperparameter learning. We compare VI, EP, Laplace approximation, and our proposed training procedure and empirically demonstrate the effectiveness of our proposal across a wide range of data sets.

Advancements in 3D Gaussian Splatting have significantly accelerated 3D reconstruction and generation. However, it may require a large number of Gaussians, which creates a substantial memory footprint. This paper introduces GES (Generalized Exponential Splatting), a novel representation that employs Generalized Exponential Function (GEF) to model 3D scenes, requiring far fewer particles to represent a scene and thus significantly outperforming Gaussian Splatting methods in efficiency with a plug-and-play replacement ability for Gaussian-based utilities. GES is validated theoretically and empirically in both principled 1D setup and realistic 3D scenes. It is shown to represent signals with sharp edges more accurately, which are typically challenging for Gaussians due to their inherent low-pass characteristics. Our empirical analysis demonstrates that GEF outperforms Gaussians in fitting natural-occurring signals (e.g. squares, triangles, and parabolic signals), thereby reducing the need for extensive splitting operations that increase the memory footprint of Gaussian Splatting. With the aid of a frequency-modulated loss, GES achieves competitive performance in novel-view synthesis benchmarks while requiring less than half the memory storage of Gaussian Splatting and increasing the rendering speed by up to 39%. The code is available on the project website https://abdullahamdi.com/ges .

Recently, impressive results have been achieved in 3D scene editing with text instructions based on a 2D diffusion model. However, current diffusion models primarily generate images by predicting noise in the latent space, and the editing is usually applied to the whole image, which makes it challenging to perform delicate, especially localized, editing for 3D scenes. Inspired by recent 3D Gaussian splatting, we propose a systematic framework, named GaussianEditor, to edit 3D scenes delicately via 3D Gaussians with text instructions. Benefiting from the explicit property of 3D Gaussians, we design a series of techniques to achieve delicate editing. Specifically, we first extract the region of interest (RoI) corresponding to the text instruction, aligning it to 3D Gaussians. The Gaussian RoI is further used to control the editing process. Our framework can achieve more delicate and precise editing of 3D scenes than previous methods while enjoying much faster training speed, i.e. within 20 minutes on a single V100 GPU, more than twice as fast as Instruct-NeRF2NeRF (45 minutes -- 2 hours).

3D Gaussian Splatting is a new method for modeling and rendering 3D radiance fields that achieves much faster learning and rendering time compared to SOTA NeRF methods. However, it comes with a drawback in the much larger storage demand compared to NeRF methods since it needs to store the parameters for several 3D Gaussians. We notice that many Gaussians may share similar parameters, so we introduce a simple vector quantization method based on \kmeans algorithm to quantize the Gaussian parameters. Then, we store the small codebook along with the index of the code for each Gaussian. Moreover, we compress the indices further by sorting them and using a method similar to run-length encoding. We do extensive experiments on standard benchmarks as well as a new benchmark which is an order of magnitude larger than the standard benchmarks. We show that our simple yet effective method can reduce the storage cost for the original 3D Gaussian Splatting method by a factor of almost 20times with a very small drop in the quality of rendered images.

We propose two approaches to extend the notion of knowledge distillation to Gaussian Process Regression (GPR) and Gaussian Process Classification (GPC); data-centric and distribution-centric. The data-centric approach resembles most current distillation techniques for machine learning, and refits a model on deterministic predictions from the teacher, while the distribution-centric approach, re-uses the full probabilistic posterior for the next iteration. By analyzing the properties of these approaches, we show that the data-centric approach for GPR closely relates to known results for self-distillation of kernel ridge regression and that the distribution-centric approach for GPR corresponds to ordinary GPR with a very particular choice of hyperparameters. Furthermore, we demonstrate that the distribution-centric approach for GPC approximately corresponds to data duplication and a particular scaling of the covariance and that the data-centric approach for GPC requires redefining the model from a Binomial likelihood to a continuous Bernoulli likelihood to be well-specified. To the best of our knowledge, our proposed approaches are the first to formulate knowledge distillation specifically for Gaussian Process models.

The surging demand for cloud computing resources, driven by the rapid growth of sophisticated large-scale models and data centers, underscores the critical importance of efficient and adaptive resource allocation. As major tech enterprises deploy massive infrastructures with thousands of GPUs, existing cloud platforms still struggle with low resource utilization due to key challenges: capturing hierarchical indicator structures, modeling non-Gaussian distributions, and decision-making under uncertainty. To address these challenges, we propose HRAMONY, an adaptive Hierarchical Attention-based Resource Modeling and Decision-Making System. HARMONY combines hierarchical multi-indicator distribution forecasting and uncertainty-aware Bayesian decision-making. It introduces a novel hierarchical attention mechanism that comprehensively models complex inter-indicator dependencies, enabling accurate predictions that can adapt to evolving environment states. By transforming Gaussian projections into adaptive non-Gaussian distributions via Normalizing Flows. Crucially, HARMONY leverages the full predictive distributions in an adaptive Bayesian process, proactively incorporating uncertainties to optimize resource allocation while robustly meeting SLA constraints under varying conditions. Extensive evaluations across four large-scale cloud datasets demonstrate HARMONY's state-of-the-art performance, significantly outperforming nine established methods. A month-long real-world deployment validated HARMONY's substantial practical impact, realizing over 35,000 GPU hours in savings and translating to $100K+ in cost reduction, showcasing its remarkable economic value through adaptive, uncertainty-aware scaling. Our code is available at https://github.com/Floating-LY/HARMONY1.

The successes of modern deep machine learning methods are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, standard theoretical approaches (formally NNGPs) involving infinite width limits eliminate representation learning. We therefore develop a new infinite width limit, the Bayesian representation learning limit, that exhibits representation learning mirroring that in finite-width models, yet at the same time, retains some of the simplicity of standard infinite-width limits. In particular, we show that Deep Gaussian processes (DGPs) in the Bayesian representation learning limit have exactly multivariate Gaussian posteriors, and the posterior covariances can be obtained by optimizing an interpretable objective combining a log-likelihood to improve performance with a series of KL-divergences which keep the posteriors close to the prior. We confirm these results experimentally in wide but finite DGPs. Next, we introduce the possibility of using this limit and objective as a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). Like most naive kernel methods, DKMs scale cubically in the number of datapoints. We therefore use methods from the Gaussian process inducing point literature to develop a sparse DKM that scales linearly in the number of datapoints. Finally, we extend these approaches to NNs (which have non-Gaussian posteriors) in the Appendices.

We propose Gaussian Frosting, a novel mesh-based representation for high-quality rendering and editing of complex 3D effects in real-time. Our approach builds on the recent 3D Gaussian Splatting framework, which optimizes a set of 3D Gaussians to approximate a radiance field from images. We propose first extracting a base mesh from Gaussians during optimization, then building and refining an adaptive layer of Gaussians with a variable thickness around the mesh to better capture the fine details and volumetric effects near the surface, such as hair or grass. We call this layer Gaussian Frosting, as it resembles a coating of frosting on a cake. The fuzzier the material, the thicker the frosting. We also introduce a parameterization of the Gaussians to enforce them to stay inside the frosting layer and automatically adjust their parameters when deforming, rescaling, editing or animating the mesh. Our representation allows for efficient rendering using Gaussian splatting, as well as editing and animation by modifying the base mesh. We demonstrate the effectiveness of our method on various synthetic and real scenes, and show that it outperforms existing surface-based approaches. We will release our code and a web-based viewer as additional contributions. Our project page is the following: https://anttwo.github.io/frosting/

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $sum_{i=1}^k w_i N(mu_i,sigma^2), i.e. the sample is observed only if it lies inside a set S. The goal is to learn the weights w_i and the means \mu_i. We propose an algorithm that takes only 1{\varepsilon^{O(k)}} samples to estimate the weights w_i and the means \mu_i within \varepsilon$ error.

Neural rendering methods have significantly advanced photo-realistic 3D scene rendering in various academic and industrial applications. The recent 3D Gaussian Splatting method has achieved the state-of-the-art rendering quality and speed combining the benefits of both primitive-based representations and volumetric representations. However, it often leads to heavily redundant Gaussians that try to fit every training view, neglecting the underlying scene geometry. Consequently, the resulting model becomes less robust to significant view changes, texture-less area and lighting effects. We introduce Scaffold-GS, which uses anchor points to distribute local 3D Gaussians, and predicts their attributes on-the-fly based on viewing direction and distance within the view frustum. Anchor growing and pruning strategies are developed based on the importance of neural Gaussians to reliably improve the scene coverage. We show that our method effectively reduces redundant Gaussians while delivering high-quality rendering. We also demonstrates an enhanced capability to accommodate scenes with varying levels-of-detail and view-dependent observations, without sacrificing the rendering speed.

In recent years, 3D Gaussian splatting has emerged as a powerful technique for 3D reconstruction and generation, known for its fast and high-quality rendering capabilities. To address these shortcomings, this paper introduces a novel diffusion-based framework, GVGEN, designed to efficiently generate 3D Gaussian representations from text input. We propose two innovative techniques:(1) Structured Volumetric Representation. We first arrange disorganized 3D Gaussian points as a structured form GaussianVolume. This transformation allows the capture of intricate texture details within a volume composed of a fixed number of Gaussians. To better optimize the representation of these details, we propose a unique pruning and densifying method named the Candidate Pool Strategy, enhancing detail fidelity through selective optimization. (2) Coarse-to-fine Generation Pipeline. To simplify the generation of GaussianVolume and empower the model to generate instances with detailed 3D geometry, we propose a coarse-to-fine pipeline. It initially constructs a basic geometric structure, followed by the prediction of complete Gaussian attributes. Our framework, GVGEN, demonstrates superior performance in qualitative and quantitative assessments compared to existing 3D generation methods. Simultaneously, it maintains a fast generation speed (sim7 seconds), effectively striking a balance between quality and efficiency.

Understanding the parameter estimation of softmax gating Gaussian mixture of experts has remained a long-standing open problem in the literature. It is mainly due to three fundamental theoretical challenges associated with the softmax gating function: (i) the identifiability only up to the translation of parameters; (ii) the intrinsic interaction via partial differential equations between the softmax gating and the expert functions in the Gaussian density; (iii) the complex dependence between the numerator and denominator of the conditional density of softmax gating Gaussian mixture of experts. We resolve these challenges by proposing novel Voronoi loss functions among parameters and establishing the convergence rates of maximum likelihood estimator (MLE) for solving parameter estimation in these models. When the true number of experts is unknown and over-specified, our findings show a connection between the convergence rate of the MLE and a solvability problem of a system of polynomial equations.

Recently, 3D Gaussian Splatting, as a novel 3D representation, has garnered attention for its fast rendering speed and high rendering quality. However, this comes with high memory consumption, e.g., a well-trained Gaussian field may utilize three million Gaussian primitives and over 700 MB of memory. We credit this high memory footprint to the lack of consideration for the relationship between primitives. In this paper, we propose a memory-efficient Gaussian field named SUNDAE with spectral pruning and neural compensation. On one hand, we construct a graph on the set of Gaussian primitives to model their relationship and design a spectral down-sampling module to prune out primitives while preserving desired signals. On the other hand, to compensate for the quality loss of pruning Gaussians, we exploit a lightweight neural network head to mix splatted features, which effectively compensates for quality losses while capturing the relationship between primitives in its weights. We demonstrate the performance of SUNDAE with extensive results. For example, SUNDAE can achieve 26.80 PSNR at 145 FPS using 104 MB memory while the vanilla Gaussian splatting algorithm achieves 25.60 PSNR at 160 FPS using 523 MB memory, on the Mip-NeRF360 dataset. Codes are publicly available at https://runyiyang.github.io/projects/SUNDAE/.

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

There has been a recent surge of interest in modeling neural networks (NNs) as Gaussian processes. In the limit of a NN of infinite width the NN becomes equivalent to a Gaussian process. Here we demonstrate that for an ensemble of large, finite, fully connected networks with a single hidden layer the distribution of outputs at initialization is well described by a Gaussian perturbed by the fourth Hermite polynomial for weights drawn from a symmetric distribution. We show that the scale of the perturbation is inversely proportional to the number of units in the NN and that higher order terms decay more rapidly, thereby recovering the Edgeworth expansion. We conclude by observing that understanding how this perturbation changes under training would reveal the regimes in which the Gaussian process framework is valid to model NN behavior.

With their meaningful geometry and their omnipresence in the 3D world, edges are extremely useful primitives in computer vision. 3D edges comprise of lines and curves, and methods to reconstruct them use either multi-view images or point clouds as input. State-of-the-art image-based methods first learn a 3D edge point cloud then fit 3D edges to it. The edge point cloud is obtained by learning a 3D neural implicit edge field from which the 3D edge points are sampled on a specific level set (0 or 1). However, such methods present two important drawbacks: i) it is not realistic to sample points on exact level sets due to float imprecision and training inaccuracies. Instead, they are sampled within a range of levels so the points do not lie accurately on the 3D edges and require further processing. ii) Such implicit representations are computationally expensive and require long training times. In this paper, we address these two limitations and propose a 3D edge mapping that is simpler, more efficient, and preserves accuracy. Our method learns explicitly the 3D edge points and their edge direction hence bypassing the need for point sampling. It casts a 3D edge point as the center of a 3D Gaussian and the edge direction as the principal axis of the Gaussian. Such a representation has the advantage of being not only geometrically meaningful but also compatible with the efficient training optimization defined in Gaussian Splatting. Results show that the proposed method produces edges as accurate and complete as the state-of-the-art while being an order of magnitude faster. Code is released at https://github.com/kunalchelani/EdgeGaussians.

Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (\ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .

In recent times, the generation of 3D assets from text prompts has shown impressive results. Both 2D and 3D diffusion models can generate decent 3D objects based on prompts. 3D diffusion models have good 3D consistency, but their quality and generalization are limited as trainable 3D data is expensive and hard to obtain. 2D diffusion models enjoy strong abilities of generalization and fine generation, but the 3D consistency is hard to guarantee. This paper attempts to bridge the power from the two types of diffusion models via the recent explicit and efficient 3D Gaussian splatting representation. A fast 3D generation framework, named as \name, is proposed, where the 3D diffusion model provides point cloud priors for initialization and the 2D diffusion model enriches the geometry and appearance. Operations of noisy point growing and color perturbation are introduced to enhance the initialized Gaussians. Our \name can generate a high-quality 3D instance within 25 minutes on one GPU, much faster than previous methods, while the generated instances can be directly rendered in real time. Demos and code are available at https://taoranyi.com/gaussiandreamer/.

Given a stationary state-space model that relates a sequence of hidden states and corresponding measurements or observations, Bayesian filtering provides a principled statistical framework for inferring the posterior distribution of the current state given all measurements up to the present time. For example, the Apollo lunar module implemented a Kalman filter to infer its location from a sequence of earth-based radar measurements and land safely on the moon. To perform Bayesian filtering, we require a measurement model that describes the conditional distribution of each observation given state. The Kalman filter takes this measurement model to be linear, Gaussian. Here we show how a nonlinear, Gaussian approximation to the distribution of state given observation can be used in conjunction with Bayes' rule to build a nonlinear, non-Gaussian measurement model. The resulting approach, called the Discriminative Kalman Filter (DKF), retains fast closed-form updates for the posterior. We argue there are many cases where the distribution of state given measurement is better-approximated as Gaussian, especially when the dimensionality of measurements far exceeds that of states and the Bernstein-von Mises theorem applies. Online neural decoding for brain-computer interfaces provides a motivating example, where filtering incorporates increasingly detailed measurements of neural activity to provide users control over external devices. Within the BrainGate2 clinical trial, the DKF successfully enabled three volunteers with quadriplegia to control an on-screen cursor in real-time using mental imagery alone. Participant "T9" used the DKF to type out messages on a tablet PC.

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

Forecasting on sparse multivariate time series (MTS) aims to model the predictors of future values of time series given their incomplete past, which is important for many emerging applications. However, most existing methods process MTS's individually, and do not leverage the dynamic distributions underlying the MTS's, leading to sub-optimal results when the sparsity is high. To address this challenge, we propose a novel generative model, which tracks the transition of latent clusters, instead of isolated feature representations, to achieve robust modeling. It is characterized by a newly designed dynamic Gaussian mixture distribution, which captures the dynamics of clustering structures, and is used for emitting timeseries. The generative model is parameterized by neural networks. A structured inference network is also designed for enabling inductive analysis. A gating mechanism is further introduced to dynamically tune the Gaussian mixture distributions. Extensive experimental results on a variety of real-life datasets demonstrate the effectiveness of our method.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

The field of novel-view synthesis has recently witnessed the emergence of 3D Gaussian Splatting, which represents scenes in a point-based manner and renders through rasterization. This methodology, in contrast to Radiance Fields that rely on ray tracing, demonstrates superior rendering quality and speed. However, the explicit and unstructured nature of 3D Gaussians poses a significant storage challenge, impeding its broader application. To address this challenge, we introduce the Gaussian-Forest modeling framework, which hierarchically represents a scene as a forest of hybrid 3D Gaussians. Each hybrid Gaussian retains its unique explicit attributes while sharing implicit ones with its sibling Gaussians, thus optimizing parameterization with significantly fewer variables. Moreover, adaptive growth and pruning strategies are designed, ensuring detailed representation in complex regions and a notable reduction in the number of required Gaussians. Extensive experiments demonstrate that Gaussian-Forest not only maintains comparable speed and quality but also achieves a compression rate surpassing 10 times, marking a significant advancement in efficient scene modeling. Codes will be available at https://github.com/Xian-Bei/GaussianForest.

Given the growing need for automatic 3D content creation pipelines, various 3D representations have been studied to generate 3D objects from a single image. Due to its superior rendering efficiency, 3D Gaussian splatting-based models have recently excelled in both 3D reconstruction and generation. 3D Gaussian splatting approaches for image to 3D generation are often optimization-based, requiring many computationally expensive score-distillation steps. To overcome these challenges, we introduce an Amortized Generative 3D Gaussian framework (AGG) that instantly produces 3D Gaussians from a single image, eliminating the need for per-instance optimization. Utilizing an intermediate hybrid representation, AGG decomposes the generation of 3D Gaussian locations and other appearance attributes for joint optimization. Moreover, we propose a cascaded pipeline that first generates a coarse representation of the 3D data and later upsamples it with a 3D Gaussian super-resolution module. Our method is evaluated against existing optimization-based 3D Gaussian frameworks and sampling-based pipelines utilizing other 3D representations, where AGG showcases competitive generation abilities both qualitatively and quantitatively while being several orders of magnitude faster. Project page: https://ir1d.github.io/AGG/

Given time series data, how can we answer questions like "what will happen in the future?" and "how did we get here?" These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.

Recent works have revealed that infinitely-wide feed-forward or recurrent neural networks of any architecture correspond to Gaussian processes referred to as Neural Network Gaussian Processes (NNGPs). While these works have extended the class of neural networks converging to Gaussian processes significantly, however, there has been little focus on broadening the class of stochastic processes that such neural networks converge to. In this work, inspired by the scale mixture of Gaussian random variables, we propose the scale mixture of NNGPs for which we introduce a prior distribution on the scale of the last-layer parameters. We show that simply introducing a scale prior on the last-layer parameters can turn infinitely-wide neural networks of any architecture into a richer class of stochastic processes. With certain scale priors, we obtain heavy-tailed stochastic processes, and in the case of inverse gamma priors, we recover Student's t processes. We further analyze the distributions of the neural networks initialized with our prior setting and trained with gradient descents and obtain similar results as for NNGPs. We present a practical posterior-inference algorithm for the scale mixture of NNGPs and empirically demonstrate its usefulness on regression and classification tasks. In particular, we show that in both tasks, the heavy-tailed stochastic processes obtained from our framework are robust to out-of-distribution data.

Gaussian analysis of new, high-angular-resolution interstellar 21-cm neutral hydrogen emission profile structure more clearly reveals the presence of the previously reported signature of the critical ionization velocity ({\it CIV}) of Helium (34 km s^{-1}). The present analysis includes 1496 component line widths for 178 neutral hydrogen profiles in two areas of sky at galactic latitudes around -50^circ, well away from the galactic plane. The new data considered here allow the interstellar abundance of Helium to be calculated, and the derived value of 0.095 pm 0.020 compares extremely well with the value of 0.085 for the cosmic abundance based on solar data. Although the precise mechanisms that give rise to the {\it CIV} effect in interstellar space are not yet understood, our results may provide additional motivation for further theoretical study of how the mechanism operates.

Despite their many desirable properties, Gaussian processes (GPs) are often compared unfavorably to deep neural networks (NNs) for lacking the ability to learn representations. Recent efforts to bridge the gap between GPs and deep NNs have yielded a new class of inter-domain variational GPs in which the inducing variables correspond to hidden units of a feedforward NN. In this work, we examine some practical issues associated with this approach and propose an extension that leverages the orthogonal decomposition of GPs to mitigate these limitations. In particular, we introduce spherical inter-domain features to construct more flexible data-dependent basis functions for both the principal and orthogonal components of the GP approximation and show that incorporating NN activation features under this framework not only alleviates these shortcomings but is more scalable than alternative strategies. Experiments on multiple benchmark datasets demonstrate the effectiveness of our approach.

This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture, which requires the derivation of closed-form expressions for the layer-wise post-activation population covariances. In light of our results, we investigate the interplay between architecture design and implicit regularization.

Expectation maximization (EM) algorithm is to find maximum likelihood solution for models having latent variables. A typical example is Gaussian Mixture Model (GMM) which requires Gaussian assumption, however, natural images are highly non-Gaussian so that GMM cannot be applied to perform clustering task on pixel space. To overcome such limitation, we propose a GAN based EM learning framework that can maximize the likelihood of images and estimate the latent variables with only the constraint of L-Lipschitz continuity. We call this model GAN-EM, which is a framework for image clustering, semi-supervised classification and dimensionality reduction. In M-step, we design a novel loss function for discriminator of GAN to perform maximum likelihood estimation (MLE) on data with soft class label assignments. Specifically, a conditional generator captures data distribution for K classes, and a discriminator tells whether a sample is real or fake for each class. Since our model is unsupervised, the class label of real data is regarded as latent variable, which is estimated by an additional network (E-net) in E-step. The proposed GAN-EM achieves state-of-the-art clustering and semi-supervised classification results on MNIST, SVHN and CelebA, as well as comparable quality of generated images to other recently developed generative models.

The recent Gaussian Splatting achieves high-quality and real-time novel-view synthesis of the 3D scenes. However, it is solely concentrated on the appearance and geometry modeling, while lacking in fine-grained object-level scene understanding. To address this issue, we propose Gaussian Grouping, which extends Gaussian Splatting to jointly reconstruct and segment anything in open-world 3D scenes. We augment each Gaussian with a compact Identity Encoding, allowing the Gaussians to be grouped according to their object instance or stuff membership in the 3D scene. Instead of resorting to expensive 3D labels, we supervise the Identity Encodings during the differentiable rendering by leveraging the 2D mask predictions by SAM, along with introduced 3D spatial consistency regularization. Comparing to the implicit NeRF representation, we show that the discrete and grouped 3D Gaussians can reconstruct, segment and edit anything in 3D with high visual quality, fine granularity and efficiency. Based on Gaussian Grouping, we further propose a local Gaussian Editing scheme, which shows efficacy in versatile scene editing applications, including 3D object removal, inpainting, colorization and scene recomposition. Our code and models will be at https://github.com/lkeab/gaussian-grouping.

3D Gaussian Splatting has recently emerged as a highly promising technique for modeling of static 3D scenes. In contrast to Neural Radiance Fields, it utilizes efficient rasterization allowing for very fast rendering at high-quality. However, the storage size is significantly higher, which hinders practical deployment, e.g.~on resource constrained devices. In this paper, we introduce a compact scene representation organizing the parameters of 3D Gaussian Splatting (3DGS) into a 2D grid with local homogeneity, ensuring a drastic reduction in storage requirements without compromising visual quality during rendering. Central to our idea is the explicit exploitation of perceptual redundancies present in natural scenes. In essence, the inherent nature of a scene allows for numerous permutations of Gaussian parameters to equivalently represent it. To this end, we propose a novel highly parallel algorithm that regularly arranges the high-dimensional Gaussian parameters into a 2D grid while preserving their neighborhood structure. During training, we further enforce local smoothness between the sorted parameters in the grid. The uncompressed Gaussians use the same structure as 3DGS, ensuring a seamless integration with established renderers. Our method achieves a reduction factor of 8x to 26x in size for complex scenes with no increase in training time, marking a substantial leap forward in the domain of 3D scene distribution and consumption. Additional information can be found on our project page: https://fraunhoferhhi.github.io/Self-Organizing-Gaussians/

Neural implicit representations, including Neural Distance Fields and Neural Radiance Fields, have demonstrated significant capabilities for reconstructing surfaces with complicated geometry and topology, and generating novel views of a scene. Nevertheless, it is challenging for users to directly deform or manipulate these implicit representations with large deformations in the real-time fashion. Gaussian Splatting(GS) has recently become a promising method with explicit geometry for representing static scenes and facilitating high-quality and real-time synthesis of novel views. However,it cannot be easily deformed due to the use of discrete Gaussians and lack of explicit topology. To address this, we develop a novel GS-based method that enables interactive deformation. Our key idea is to design an innovative mesh-based GS representation, which is integrated into Gaussian learning and manipulation. 3D Gaussians are defined over an explicit mesh, and they are bound with each other: the rendering of 3D Gaussians guides the mesh face split for adaptive refinement, and the mesh face split directs the splitting of 3D Gaussians. Moreover, the explicit mesh constraints help regularize the Gaussian distribution, suppressing poor-quality Gaussians(e.g. misaligned Gaussians,long-narrow shaped Gaussians), thus enhancing visual quality and avoiding artifacts during deformation. Based on this representation, we further introduce a large-scale Gaussian deformation technique to enable deformable GS, which alters the parameters of 3D Gaussians according to the manipulation of the associated mesh. Our method benefits from existing mesh deformation datasets for more realistic data-driven Gaussian deformation. Extensive experiments show that our approach achieves high-quality reconstruction and effective deformation, while maintaining the promising rendering results at a high frame rate(65 FPS on average).

Rendering dynamic scenes from monocular videos is a crucial yet challenging task. The recent deformable Gaussian Splatting has emerged as a robust solution to represent real-world dynamic scenes. However, it often leads to heavily redundant Gaussians, attempting to fit every training view at various time steps, leading to slower rendering speeds. Additionally, the attributes of Gaussians in static areas are time-invariant, making it unnecessary to model every Gaussian, which can cause jittering in static regions. In practice, the primary bottleneck in rendering speed for dynamic scenes is the number of Gaussians. In response, we introduce Efficient Dynamic Gaussian Splatting (EDGS), which represents dynamic scenes via sparse time-variant attribute modeling. Our approach formulates dynamic scenes using a sparse anchor-grid representation, with the motion flow of dense Gaussians calculated via a classical kernel representation. Furthermore, we propose an unsupervised strategy to efficiently filter out anchors corresponding to static areas. Only anchors associated with deformable objects are input into MLPs to query time-variant attributes. Experiments on two real-world datasets demonstrate that our EDGS significantly improves the rendering speed with superior rendering quality compared to previous state-of-the-art methods.

Recently, a range of neural network-based methods for image rendering have been introduced. One such widely-researched neural radiance field (NeRF) relies on a neural network to represent 3D scenes, allowing for realistic view synthesis from a small number of 2D images. However, most NeRF models are constrained by long training and inference times. In comparison, Gaussian Splatting (GS) is a novel, state-of-the-art technique for rendering points in a 3D scene by approximating their contribution to image pixels through Gaussian distributions, warranting fast training and swift, real-time rendering. A drawback of GS is the absence of a well-defined approach for its conditioning due to the necessity to condition several hundred thousand Gaussian components. To solve this, we introduce the Gaussian Mesh Splatting (GaMeS) model, which allows modification of Gaussian components in a similar way as meshes. We parameterize each Gaussian component by the vertices of the mesh face. Furthermore, our model needs mesh initialization on input or estimated mesh during training. We also define Gaussian splats solely based on their location on the mesh, allowing for automatic adjustments in position, scale, and rotation during animation. As a result, we obtain a real-time rendering of editable GS.

We introduce a parametric nonlinear transformation that is well-suited for Gaussianizing data from natural images. The data are linearly transformed, and each component is then normalized by a pooled activity measure, computed by exponentiating a weighted sum of rectified and exponentiated components and a constant. We optimize the parameters of the full transformation (linear transform, exponents, weights, constant) over a database of natural images, directly minimizing the negentropy of the responses. The optimized transformation substantially Gaussianizes the data, achieving a significantly smaller mutual information between transformed components than alternative methods including ICA and radial Gaussianization. The transformation is differentiable and can be efficiently inverted, and thus induces a density model on images. We show that samples of this model are visually similar to samples of natural image patches. We demonstrate the use of the model as a prior probability density that can be used to remove additive noise. Finally, we show that the transformation can be cascaded, with each layer optimized using the same Gaussianization objective, thus offering an unsupervised method of optimizing a deep network architecture.

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) infinite-width BNNs are particularly promising, especially in high dimensions.

The binary sum-of-digits function s returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers tgeq 0, the set of nonnegative integers n such that s(n+t)geq s(n) has asymptotic density strictly larger than 1/2. We are concerned with the block-additive function r returning the number of (overlapping) occurrences of the block 11 in the binary expansion of n. The main result of this paper is a central limit-type theorem for the difference r(n+t)-r(n): the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to 0 as the number of blocks of ones in the binary expansion of t tends to infty.

We study the problem of privately estimating the parameters of d-dimensional Gaussian Mixture Models (GMMs) with k components. For this, we develop a technique to reduce the problem to its non-private counterpart. This allows us to privatize existing non-private algorithms in a blackbox manner, while incurring only a small overhead in the sample complexity and running time. As the main application of our framework, we develop an (varepsilon, delta)-differentially private algorithm to learn GMMs using the non-private algorithm of Moitra and Valiant [MV10] as a blackbox. Consequently, this gives the first sample complexity upper bound and first polynomial time algorithm for privately learning GMMs without any boundedness assumptions on the parameters. As part of our analysis, we prove a tight (up to a constant factor) lower bound on the total variation distance of high-dimensional Gaussians which can be of independent interest.

We demonstrate the feasibility of integrating physics-based animations of solids and fluids with 3D Gaussian Splatting (3DGS) to create novel effects in virtual scenes reconstructed using 3DGS. Leveraging the coherence of the Gaussian splatting and position-based dynamics (PBD) in the underlying representation, we manage rendering, view synthesis, and the dynamics of solids and fluids in a cohesive manner. Similar to Gaussian shader, we enhance each Gaussian kernel with an added normal, aligning the kernel's orientation with the surface normal to refine the PBD simulation. This approach effectively eliminates spiky noises that arise from rotational deformation in solids. It also allows us to integrate physically based rendering to augment the dynamic surface reflections on fluids. Consequently, our framework is capable of realistically reproducing surface highlights on dynamic fluids and facilitating interactions between scene objects and fluids from new views. For more information, please visit our project page at https://amysteriouscat.github.io/GaussianSplashing/.

We propose a method to allow precise and extremely fast mesh extraction from 3D Gaussian Splatting. Gaussian Splatting has recently become very popular as it yields realistic rendering while being significantly faster to train than NeRFs. It is however challenging to extract a mesh from the millions of tiny 3D gaussians as these gaussians tend to be unorganized after optimization and no method has been proposed so far. Our first key contribution is a regularization term that encourages the gaussians to align well with the surface of the scene. We then introduce a method that exploits this alignment to extract a mesh from the Gaussians using Poisson reconstruction, which is fast, scalable, and preserves details, in contrast to the Marching Cubes algorithm usually applied to extract meshes from Neural SDFs. Finally, we introduce an optional refinement strategy that binds gaussians to the surface of the mesh, and jointly optimizes these Gaussians and the mesh through Gaussian splatting rendering. This enables easy editing, sculpting, rigging, animating, compositing and relighting of the Gaussians using traditional softwares by manipulating the mesh instead of the gaussians themselves. Retrieving such an editable mesh for realistic rendering is done within minutes with our method, compared to hours with the state-of-the-art methods on neural SDFs, while providing a better rendering quality.

Novel view synthesis has advanced significantly with the development of neural radiance fields (NeRF) and 3D Gaussian splatting (3DGS). However, achieving high quality without compromising real-time rendering remains challenging, particularly for physically-based ray tracing with view-dependent effects. Recently, N-dimensional Gaussians (N-DG) introduced a 6D spatial-angular representation to better incorporate view-dependent effects, but the Gaussian representation and control scheme are sub-optimal. In this paper, we revisit 6D Gaussians and introduce 6D Gaussian Splatting (6DGS), which enhances color and opacity representations and leverages the additional directional information in the 6D space for optimized Gaussian control. Our approach is fully compatible with the 3DGS framework and significantly improves real-time radiance field rendering by better modeling view-dependent effects and fine details. Experiments demonstrate that 6DGS significantly outperforms 3DGS and N-DG, achieving up to a 15.73 dB improvement in PSNR with a reduction of 66.5% Gaussian points compared to 3DGS. The project page is: https://gaozhongpai.github.io/6dgs/

Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper is devoted to the study of the problem of model selection among a collection of GLoME or BLoME models characterized by the number of mixture components, the complexity of Gaussian mean experts, and the hidden block-diagonal structures of the covariance matrices, in a penalized maximum likelihood estimation framework. In particular, we establish non-asymptotic risk bounds that take the form of weak oracle inequalities, provided that lower bounds for the penalties hold. The good empirical behavior of our models is then demonstrated on synthetic and real datasets.

We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by 1/depth (the only nontrivial scaling), result in the same covariance structure no matter how that limit is taken. This explains why the standard infinite-width-then-depth approach provides practical insights even for networks with depth of the same order as width. We also demonstrate that the pre-activations, in this case, have Gaussian distributions which has direct applications in Bayesian deep learning. We conduct extensive simulations that show an excellent match with our theoretical findings.

Mixture-of-experts (MoE) model incorporates the power of multiple submodels via gating functions to achieve greater performance in numerous regression and classification applications. From a theoretical perspective, while there have been previous attempts to comprehend the behavior of that model under the regression settings through the convergence analysis of maximum likelihood estimation in the Gaussian MoE model, such analysis under the setting of a classification problem has remained missing in the literature. We close this gap by establishing the convergence rates of density estimation and parameter estimation in the softmax gating multinomial logistic MoE model. Notably, when part of the expert parameters vanish, these rates are shown to be slower than polynomial rates owing to an inherent interaction between the softmax gating and expert functions via partial differential equations. To address this issue, we propose using a novel class of modified softmax gating functions which transform the input value before delivering them to the gating functions. As a result, the previous interaction disappears and the parameter estimation rates are significantly improved.

In crowd counting, each training image contains multiple people, where each person is annotated by a dot. Existing crowd counting methods need to use a Gaussian to smooth each annotated dot or to estimate the likelihood of every pixel given the annotated point. In this paper, we show that imposing Gaussians to annotations hurts generalization performance. Instead, we propose to use Distribution Matching for crowd COUNTing (DM-Count). In DM-Count, we use Optimal Transport (OT) to measure the similarity between the normalized predicted density map and the normalized ground truth density map. To stabilize OT computation, we include a Total Variation loss in our model. We show that the generalization error bound of DM-Count is tighter than that of the Gaussian smoothed methods. In terms of Mean Absolute Error, DM-Count outperforms the previous state-of-the-art methods by a large margin on two large-scale counting datasets, UCF-QNRF and NWPU, and achieves the state-of-the-art results on the ShanghaiTech and UCF-CC50 datasets. DM-Count reduced the error of the state-of-the-art published result by approximately 16%. Code is available at https://github.com/cvlab-stonybrook/DM-Count.

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

Interactive segmentation of 3D Gaussians opens a great opportunity for real-time manipulation of 3D scenes thanks to the real-time rendering capability of 3D Gaussian Splatting. However, the current methods suffer from time-consuming post-processing to deal with noisy segmentation output. Also, they struggle to provide detailed segmentation, which is important for fine-grained manipulation of 3D scenes. In this study, we propose Click-Gaussian, which learns distinguishable feature fields of two-level granularity, facilitating segmentation without time-consuming post-processing. We delve into challenges stemming from inconsistently learned feature fields resulting from 2D segmentation obtained independently from a 3D scene. 3D segmentation accuracy deteriorates when 2D segmentation results across the views, primary cues for 3D segmentation, are in conflict. To overcome these issues, we propose Global Feature-guided Learning (GFL). GFL constructs the clusters of global feature candidates from noisy 2D segments across the views, which smooths out noises when training the features of 3D Gaussians. Our method runs in 10 ms per click, 15 to 130 times as fast as the previous methods, while also significantly improving segmentation accuracy. Our project page is available at https://seokhunchoi.github.io/Click-Gaussian

A desired closure property in Bayesian probability is that an updated posterior distribution be in the same class of distributions --- say Gaussians --- as the prior distribution. When the updating takes place via a statistical model, one calls the class of prior distributions the `conjugate priors' of the model. This paper gives (1) an abstract formulation of this notion of conjugate prior, using channels, in a graphical language, (2) a simple abstract proof that such conjugate priors yield Bayesian inversions, and (3) a logical description of conjugate priors that highlights the required closure of the priors under updating. The theory is illustrated with several standard examples, also covering multiple updating.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Implicit neural representations (INRs) recently achieved great success in image representation and compression, offering high visual quality and fast rendering speeds with 10-1000 FPS, assuming sufficient GPU resources are available. However, this requirement often hinders their use on low-end devices with limited memory. In response, we propose a groundbreaking paradigm of image representation and compression by 2D Gaussian Splatting, named GaussianImage. We first introduce 2D Gaussian to represent the image, where each Gaussian has 8 parameters including position, covariance and color. Subsequently, we unveil a novel rendering algorithm based on accumulated summation. Remarkably, our method with a minimum of 3times lower GPU memory usage and 5times faster fitting time not only rivals INRs (e.g., WIRE, I-NGP) in representation performance, but also delivers a faster rendering speed of 1500-2000 FPS regardless of parameter size. Furthermore, we integrate existing vector quantization technique to build an image codec. Experimental results demonstrate that our codec attains rate-distortion performance comparable to compression-based INRs such as COIN and COIN++, while facilitating decoding speeds of approximately 1000 FPS. Additionally, preliminary proof of concept shows that our codec surpasses COIN and COIN++ in performance when using partial bits-back coding.

The increasing demand for virtual reality applications has highlighted the significance of crafting immersive 3D assets. We present a text-to-3D 360^{circ} scene generation pipeline that facilitates the creation of comprehensive 360^{circ} scenes for in-the-wild environments in a matter of minutes. Our approach utilizes the generative power of a 2D diffusion model and prompt self-refinement to create a high-quality and globally coherent panoramic image. This image acts as a preliminary "flat" (2D) scene representation. Subsequently, it is lifted into 3D Gaussians, employing splatting techniques to enable real-time exploration. To produce consistent 3D geometry, our pipeline constructs a spatially coherent structure by aligning the 2D monocular depth into a globally optimized point cloud. This point cloud serves as the initial state for the centroids of 3D Gaussians. In order to address invisible issues inherent in single-view inputs, we impose semantic and geometric constraints on both synthesized and input camera views as regularizations. These guide the optimization of Gaussians, aiding in the reconstruction of unseen regions. In summary, our method offers a globally consistent 3D scene within a 360^{circ} perspective, providing an enhanced immersive experience over existing techniques. Project website at: http://dreamscene360.github.io/

This paper proposes a novel method for deep learning based on the analytical convolution of multidimensional Gaussian mixtures. In contrast to tensors, these do not suffer from the curse of dimensionality and allow for a compact representation, as data is only stored where details exist. Convolution kernels and data are Gaussian mixtures with unconstrained weights, positions, and covariance matrices. Similar to discrete convolutional networks, each convolution step produces several feature channels, represented by independent Gaussian mixtures. Since traditional transfer functions like ReLUs do not produce Gaussian mixtures, we propose using a fitting of these functions instead. This fitting step also acts as a pooling layer if the number of Gaussian components is reduced appropriately. We demonstrate that networks based on this architecture reach competitive accuracy on Gaussian mixtures fitted to the MNIST and ModelNet data sets.

Constructing a 3D scene capable of accommodating open-ended language queries, is a pivotal pursuit, particularly within the domain of robotics. Such technology facilitates robots in executing object manipulations based on human language directives. To tackle this challenge, some research efforts have been dedicated to the development of language-embedded implicit fields. However, implicit fields (e.g. NeRF) encounter limitations due to the necessity of processing a large number of input views for reconstruction, coupled with their inherent inefficiencies in inference. Thus, we present the GaussianGrasper, which utilizes 3D Gaussian Splatting to explicitly represent the scene as a collection of Gaussian primitives. Our approach takes a limited set of RGB-D views and employs a tile-based splatting technique to create a feature field. In particular, we propose an Efficient Feature Distillation (EFD) module that employs contrastive learning to efficiently and accurately distill language embeddings derived from foundational models. With the reconstructed geometry of the Gaussian field, our method enables the pre-trained grasping model to generate collision-free grasp pose candidates. Furthermore, we propose a normal-guided grasp module to select the best grasp pose. Through comprehensive real-world experiments, we demonstrate that GaussianGrasper enables robots to accurately query and grasp objects with language instructions, providing a new solution for language-guided manipulation tasks. Data and codes can be available at https://github.com/MrSecant/GaussianGrasper.

Mixture of experts (MoE) has a well-principled finite mixture model construction for prediction, allowing the gating network (mixture weights) to learn from the predictors (explanatory variables) together with the experts' network (mixture component densities). We investigate the estimation properties of MoEs in a high-dimensional setting, where the number of predictors is much larger than the sample size, for which the literature lacks computational and especially theoretical results. We consider the class of finite MoE models with softmax gating functions and Gaussian regression experts, and focus on the theoretical properties of their l_1-regularized estimation via the Lasso. We provide a lower bound on the regularization parameter of the Lasso penalty that ensures an l_1-oracle inequality is satisfied by the Lasso estimator according to the Kullback--Leibler loss. We further state an l_1-ball oracle inequality for the l_1-penalized maximum likelihood estimator from the model selection.

Standard infinite-width limits of neural networks sacrifice the ability for intermediate layers to learn representations from data. Recent work (A theory of representation learning gives a deep generalisation of kernel methods, Yang et al. 2023) modified the Neural Network Gaussian Process (NNGP) limit of Bayesian neural networks so that representation learning is retained. Furthermore, they found that applying this modified limit to a deep Gaussian process gives a practical learning algorithm which they dubbed the deep kernel machine (DKM). However, they only considered the simplest possible setting: regression in small, fully connected networks with e.g. 10 input features. Here, we introduce convolutional deep kernel machines. This required us to develop a novel inter-domain inducing point approximation, as well as introducing and experimentally assessing a number of techniques not previously seen in DKMs, including analogues to batch normalisation, different likelihoods, and different types of top-layer. The resulting model trains in roughly 77 GPU hours, achieving around 99% test accuracy on MNIST, 72% on CIFAR-100, and 92.7% on CIFAR-10, which is SOTA for kernel methods.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-m rate, where m is the number of simulated samples. This can lead to significant computational challenges since a large m is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

We introduce PhysGaussian, a new method that seamlessly integrates physically grounded Newtonian dynamics within 3D Gaussians to achieve high-quality novel motion synthesis. Employing a custom Material Point Method (MPM), our approach enriches 3D Gaussian kernels with physically meaningful kinematic deformation and mechanical stress attributes, all evolved in line with continuum mechanics principles. A defining characteristic of our method is the seamless integration between physical simulation and visual rendering: both components utilize the same 3D Gaussian kernels as their discrete representations. This negates the necessity for triangle/tetrahedron meshing, marching cubes, "cage meshes," or any other geometry embedding, highlighting the principle of "what you see is what you simulate (WS^2)." Our method demonstrates exceptional versatility across a wide variety of materials--including elastic entities, metals, non-Newtonian fluids, and granular materials--showcasing its strong capabilities in creating diverse visual content with novel viewpoints and movements. Our project page is at: https://xpandora.github.io/PhysGaussian/

Despite much progress, achieving real-time high-fidelity head avatar animation is still difficult and existing methods have to trade-off between speed and quality. 3DMM based methods often fail to model non-facial structures such as eyeglasses and hairstyles, while neural implicit models suffer from deformation inflexibility and rendering inefficiency. Although 3D Gaussian has been demonstrated to possess promising capability for geometry representation and radiance field reconstruction, applying 3D Gaussian in head avatar creation remains a major challenge since it is difficult for 3D Gaussian to model the head shape variations caused by changing poses and expressions. In this paper, we introduce PSAvatar, a novel framework for animatable head avatar creation that utilizes discrete geometric primitive to create a parametric morphable shape model and employs 3D Gaussian for fine detail representation and high fidelity rendering. The parametric morphable shape model is a Point-based Morphable Shape Model (PMSM) which uses points instead of meshes for 3D representation to achieve enhanced representation flexibility. The PMSM first converts the FLAME mesh to points by sampling on the surfaces as well as off the meshes to enable the reconstruction of not only surface-like structures but also complex geometries such as eyeglasses and hairstyles. By aligning these points with the head shape in an analysis-by-synthesis manner, the PMSM makes it possible to utilize 3D Gaussian for fine detail representation and appearance modeling, thus enabling the creation of high-fidelity avatars. We show that PSAvatar can reconstruct high-fidelity head avatars of a variety of subjects and the avatars can be animated in real-time (ge 25 fps at a resolution of 512 times 512 ).

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

We present a novel stochastic variational Gaussian process (GP) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for GPs (CVTGP) is defined in terms of the GP prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent GP coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to O(M), enjoys numerical stability, and maintains O(M^3) time- and O(M^2) space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic GP inference methods.

Recent advances in radiance field reconstruction, such as 3D Gaussian Splatting (3DGS), have achieved high-quality novel view synthesis and fast rendering by representing scenes with compositions of Gaussian primitives. However, 3D Gaussians present several limitations for scene reconstruction. Accurately capturing hard edges is challenging without significantly increasing the number of Gaussians, creating a large memory footprint. Moreover, they struggle to represent flat surfaces, as they are diffused in space. Without hand-crafted regularizers, they tend to disperse irregularly around the actual surface. To circumvent these issues, we introduce a novel method, named 3D Convex Splatting (3DCS), which leverages 3D smooth convexes as primitives for modeling geometrically-meaningful radiance fields from multi-view images. Smooth convex shapes offer greater flexibility than Gaussians, allowing for a better representation of 3D scenes with hard edges and dense volumes using fewer primitives. Powered by our efficient CUDA-based rasterizer, 3DCS achieves superior performance over 3DGS on benchmarks such as Mip-NeRF360, Tanks and Temples, and Deep Blending. Specifically, our method attains an improvement of up to 0.81 in PSNR and 0.026 in LPIPS compared to 3DGS while maintaining high rendering speeds and reducing the number of required primitives. Our results highlight the potential of 3D Convex Splatting to become the new standard for high-quality scene reconstruction and novel view synthesis. Project page: convexsplatting.github.io.

The 3D Gaussian Splatting (3DGS) gained its popularity recently by combining the advantages of both primitive-based and volumetric 3D representations, resulting in improved quality and efficiency for 3D scene rendering. However, 3DGS is not alias-free, and its rendering at varying resolutions could produce severe blurring or jaggies. This is because 3DGS treats each pixel as an isolated, single point rather than as an area, causing insensitivity to changes in the footprints of pixels. Consequently, this discrete sampling scheme inevitably results in aliasing, owing to the restricted sampling bandwidth. In this paper, we derive an analytical solution to address this issue. More specifically, we use a conditioned logistic function as the analytic approximation of the cumulative distribution function (CDF) in a one-dimensional Gaussian signal and calculate the Gaussian integral by subtracting the CDFs. We then introduce this approximation in the two-dimensional pixel shading, and present Analytic-Splatting, which analytically approximates the Gaussian integral within the 2D-pixel window area to better capture the intensity response of each pixel. Moreover, we use the approximated response of the pixel window integral area to participate in the transmittance calculation of volume rendering, making Analytic-Splatting sensitive to the changes in pixel footprint at different resolutions. Experiments on various datasets validate that our approach has better anti-aliasing capability that gives more details and better fidelity.

We introduce the Splatter Image, an ultra-fast approach for monocular 3D object reconstruction which operates at 38 FPS. Splatter Image is based on Gaussian Splatting, which has recently brought real-time rendering, fast training, and excellent scaling to multi-view reconstruction. For the first time, we apply Gaussian Splatting in a monocular reconstruction setting. Our approach is learning-based, and, at test time, reconstruction only requires the feed-forward evaluation of a neural network. The main innovation of Splatter Image is the surprisingly straightforward design: it uses a 2D image-to-image network to map the input image to one 3D Gaussian per pixel. The resulting Gaussians thus have the form of an image, the Splatter Image. We further extend the method to incorporate more than one image as input, which we do by adding cross-view attention. Owning to the speed of the renderer (588 FPS), we can use a single GPU for training while generating entire images at each iteration in order to optimize perceptual metrics like LPIPS. On standard benchmarks, we demonstrate not only fast reconstruction but also better results than recent and much more expensive baselines in terms of PSNR, LPIPS, and other metrics.

Top-K sparse softmax gating mixture of experts has been widely used for scaling up massive deep-learning architectures without increasing the computational cost. Despite its popularity in real-world applications, the theoretical understanding of that gating function has remained an open problem. The main challenge comes from the structure of the top-K sparse softmax gating function, which partitions the input space into multiple regions with distinct behaviors. By focusing on a Gaussian mixture of experts, we establish theoretical results on the effects of the top-K sparse softmax gating function on both density and parameter estimations. Our results hinge upon defining novel loss functions among parameters to capture different behaviors of the input regions. When the true number of experts k_{ast} is known, we demonstrate that the convergence rates of density and parameter estimations are both parametric on the sample size. However, when k_{ast} becomes unknown and the true model is over-specified by a Gaussian mixture of k experts where k > k_{ast}, our findings suggest that the number of experts selected from the top-K sparse softmax gating function must exceed the total cardinality of a certain number of Voronoi cells associated with the true parameters to guarantee the convergence of the density estimation. Moreover, while the density estimation rate remains parametric under this setting, the parameter estimation rates become substantially slow due to an intrinsic interaction between the softmax gating and expert functions.

Novel-view synthesis is an important problem in computer vision with applications in 3D reconstruction, mixed reality, and robotics. Recent methods like 3D Gaussian Splatting (3DGS) have become the preferred method for this task, providing high-quality novel views in real time. However, the training time of a 3DGS model is slow, often taking 30 minutes for a scene with 200 views. In contrast, our goal is to reduce the optimization time by training for fewer steps while maintaining high rendering quality. Specifically, we combine the guidance from both the position error and the appearance error to achieve a more effective densification. To balance the rate between adding new Gaussians and fitting old Gaussians, we develop a convergence-aware budget control mechanism. Moreover, to make the densification process more reliable, we selectively add new Gaussians from mostly visited regions. With these designs, we reduce the Gaussian optimization steps to one-third of the previous approach while achieving a comparable or even better novel view rendering quality. To further facilitate the rapid fitting of 4K resolution images, we introduce a dilation-based rendering technique. Our method, Turbo-GS, speeds up optimization for typical scenes and scales well to high-resolution (4K) scenarios on standard datasets. Through extensive experiments, we show that our method is significantly faster in optimization than other methods while retaining quality. Project page: https://ivl.cs.brown.edu/research/turbo-gs.

The field of 3D reconstruction from images has rapidly evolved in the past few years, first with the introduction of Neural Radiance Field (NeRF) and more recently with 3D Gaussian Splatting (3DGS). The latter provides a significant edge over NeRF in terms of the training and inference speed, as well as the reconstruction quality. Although 3DGS works well for dense input images, the unstructured point-cloud like representation quickly overfits to the more challenging setup of extremely sparse input images (e.g., 3 images), creating a representation that appears as a jumble of needles from novel views. To address this issue, we propose regularized optimization and depth-based initialization. Our key idea is to introduce a structured Gaussian representation that can be controlled in 2D image space. We then constraint the Gaussians, in particular their position, and prevent them from moving independently during optimization. Specifically, we introduce single and multiview constraints through an implicit convolutional decoder and a total variation loss, respectively. With the coherency introduced to the Gaussians, we further constrain the optimization through a flow-based loss function. To support our regularized optimization, we propose an approach to initialize the Gaussians using monocular depth estimates at each input view. We demonstrate significant improvements compared to the state-of-the-art sparse-view NeRF-based approaches on a variety of scenes.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

3D Gaussian Splatting (3DGS) achieves fast and high-quality renderings by using numerous small Gaussians, which leads to significant memory consumption. This reliance on a large number of Gaussians restricts the application of 3DGS-based models on low-cost devices due to memory limitations. However, simply reducing the number of Gaussians to accommodate devices with less memory capacity leads to inferior quality compared to the quality that can be achieved on high-end hardware. To address this lack of scalability, we propose integrating a Flexible Level of Detail (FLoD) to 3DGS, to allow a scene to be rendered at varying levels of detail according to hardware capabilities. While existing 3DGSs with LoD focus on detailed reconstruction, our method provides reconstructions using a small number of Gaussians for reduced memory requirements, and a larger number of Gaussians for greater detail. Experiments demonstrate our various rendering options with tradeoffs between rendering quality and memory usage, thereby allowing real-time rendering across different memory constraints. Furthermore, we show that our method generalizes to different 3DGS frameworks, indicating its potential for integration into future state-of-the-art developments. Project page: https://3dgs-flod.github.io/flod.github.io/

Many state-of-the-art hyperparameter optimization (HPO) algorithms rely on model-based optimizers that learn surrogate models of the target function to guide the search. Gaussian processes are the de facto surrogate model due to their ability to capture uncertainty but they make strong assumptions about the observation noise, which might not be warranted in practice. In this work, we propose to leverage conformalized quantile regression which makes minimal assumptions about the observation noise and, as a result, models the target function in a more realistic and robust fashion which translates to quicker HPO convergence on empirical benchmarks. To apply our method in a multi-fidelity setting, we propose a simple, yet effective, technique that aggregates observed results across different resource levels and outperforms conventional methods across many empirical tasks.

3D Gaussian Splatting (3DGS) has recently revolutionized radiance field reconstruction, achieving high quality novel view synthesis and fast rendering speed without baking. However, 3DGS fails to accurately represent surfaces due to the multi-view inconsistent nature of 3D Gaussians. We present 2D Gaussian Splatting (2DGS), a novel approach to model and reconstruct geometrically accurate radiance fields from multi-view images. Our key idea is to collapse the 3D volume into a set of 2D oriented planar Gaussian disks. Unlike 3D Gaussians, 2D Gaussians provide view-consistent geometry while modeling surfaces intrinsically. To accurately recover thin surfaces and achieve stable optimization, we introduce a perspective-accurate 2D splatting process utilizing ray-splat intersection and rasterization. Additionally, we incorporate depth distortion and normal consistency terms to further enhance the quality of the reconstructions. We demonstrate that our differentiable renderer allows for noise-free and detailed geometry reconstruction while maintaining competitive appearance quality, fast training speed, and real-time rendering. Our code will be made publicly available.

This work introduces the Efficient Transformed Gaussian Process (ETGP), a new way of creating C stochastic processes characterized by: 1) the C processes are non-stationary, 2) the C processes are dependent by construction without needing a mixing matrix, 3) training and making predictions is very efficient since the number of Gaussian Processes (GP) operations (e.g. inverting the inducing point's covariance matrix) do not depend on the number of processes. This makes the ETGP particularly suited for multi-class problems with a very large number of classes, which are the problems studied in this work. ETGPs exploit the recently proposed Transformed Gaussian Process (TGP), a stochastic process specified by transforming a Gaussian Process using an invertible transformation. However, unlike TGPs, ETGPs are constructed by transforming a single sample from a GP using C invertible transformations. We derive an efficient sparse variational inference algorithm for the proposed model and demonstrate its utility in 5 classification tasks which include low/medium/large datasets and a different number of classes, ranging from just a few to hundreds. Our results show that ETGPs, in general, outperform state-of-the-art methods for multi-class classification based on GPs, and have a lower computational cost (around one order of magnitude smaller).

3D Gaussian Splatting (3D-GS) has recently attracted great attention with real-time and photo-realistic renderings. This technique typically takes perspective images as input and optimizes a set of 3D elliptical Gaussians by splatting them onto the image planes, resulting in 2D Gaussians. However, applying 3D-GS to panoramic inputs presents challenges in effectively modeling the projection onto the spherical surface of {360^circ} images using 2D Gaussians. In practical applications, input panoramas are often sparse, leading to unreliable initialization of 3D Gaussians and subsequent degradation of 3D-GS quality. In addition, due to the under-constrained geometry of texture-less planes (e.g., walls and floors), 3D-GS struggles to model these flat regions with elliptical Gaussians, resulting in significant floaters in novel views. To address these issues, we propose 360-GS, a novel 360^{circ} Gaussian splatting for a limited set of panoramic inputs. Instead of splatting 3D Gaussians directly onto the spherical surface, 360-GS projects them onto the tangent plane of the unit sphere and then maps them to the spherical projections. This adaptation enables the representation of the projection using Gaussians. We guide the optimization of 360-GS by exploiting layout priors within panoramas, which are simple to obtain and contain strong structural information about the indoor scene. Our experimental results demonstrate that 360-GS allows panoramic rendering and outperforms state-of-the-art methods with fewer artifacts in novel view synthesis, thus providing immersive roaming in indoor scenarios.

Interrogating the evolution of biological changes at early stages of life requires longitudinal profiling of molecules, such as DNA methylation, which can be challenging with children. We introduce a probabilistic and longitudinal machine learning framework based on multi-mean Gaussian processes (GPs), accounting for individual and gene correlations across time. This method provides future predictions of DNA methylation status at different individual ages while accounting for uncertainty. Our model is trained on a birth cohort of children with methylation profiled at ages 0-4, and we demonstrated that the status of methylation sites for each child can be accurately predicted at ages 5-7. We show that methylation profiles predicted by multi-mean GPs can be used to estimate other phenotypes, such as epigenetic age, and enable comparison to other health measures of interest. This approach encourages epigenetic studies to move towards longitudinal design for investigating epigenetic changes during development, ageing and disease progression.

Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation~(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.

We present latentSplat, a method to predict semantic Gaussians in a 3D latent space that can be splatted and decoded by a light-weight generative 2D architecture. Existing methods for generalizable 3D reconstruction either do not scale to large scenes and resolutions, or are limited to interpolation of close input views. latentSplat combines the strengths of regression-based and generative approaches while being trained purely on readily available real video data. The core of our method are variational 3D Gaussians, a representation that efficiently encodes varying uncertainty within a latent space consisting of 3D feature Gaussians. From these Gaussians, specific instances can be sampled and rendered via efficient splatting and a fast, generative decoder. We show that latentSplat outperforms previous works in reconstruction quality and generalization, while being fast and scalable to high-resolution data.

Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.

Latent Gaussian process (GP) models are widely used in neuroscience to uncover hidden state evolutions from sequential observations, mainly in neural activity recordings. While latent GP models provide a principled and powerful solution in theory, the intractable posterior in non-conjugate settings necessitates approximate inference schemes, which may lack scalability. In this work, we propose cvHM, a general inference framework for latent GP models leveraging Hida-Mat\'ern kernels and conjugate computation variational inference (CVI). With cvHM, we are able to perform variational inference of latent neural trajectories with linear time complexity for arbitrary likelihoods. The reparameterization of stationary kernels using Hida-Mat\'ern GPs helps us connect the latent variable models that encode prior assumptions through dynamical systems to those that encode trajectory assumptions through GPs. In contrast to previous work, we use bidirectional information filtering, leading to a more concise implementation. Furthermore, we employ the Whittle approximate likelihood to achieve highly efficient hyperparameter learning.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

The present work investigates two properties of level crossings of a stationary Gaussian process X(t) with autocorrelation function R_X(tau). We show firstly that if R_X(tau) admits finite second and fourth derivatives at the origin, the length of up-excursions above a large negative level -gamma is asymptotically exponential as -gamma to -infty. Secondly, assuming that R_X(tau) admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two subsequent large levels. The asymptotic results are shown to be effective even for moderate values of crossing level. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.

This paper aims to address the challenge of reconstructing long volumetric videos from multi-view RGB videos. Recent dynamic view synthesis methods leverage powerful 4D representations, like feature grids or point cloud sequences, to achieve high-quality rendering results. However, they are typically limited to short (1~2s) video clips and often suffer from large memory footprints when dealing with longer videos. To solve this issue, we propose a novel 4D representation, named Temporal Gaussian Hierarchy, to compactly model long volumetric videos. Our key observation is that there are generally various degrees of temporal redundancy in dynamic scenes, which consist of areas changing at different speeds. Motivated by this, our approach builds a multi-level hierarchy of 4D Gaussian primitives, where each level separately describes scene regions with different degrees of content change, and adaptively shares Gaussian primitives to represent unchanged scene content over different temporal segments, thus effectively reducing the number of Gaussian primitives. In addition, the tree-like structure of the Gaussian hierarchy allows us to efficiently represent the scene at a particular moment with a subset of Gaussian primitives, leading to nearly constant GPU memory usage during the training or rendering regardless of the video length. Extensive experimental results demonstrate the superiority of our method over alternative methods in terms of training cost, rendering speed, and storage usage. To our knowledge, this work is the first approach capable of efficiently handling minutes of volumetric video data while maintaining state-of-the-art rendering quality. Our project page is available at: https://zju3dv.github.io/longvolcap.

Sequential VAEs have been successfully considered for many high-dimensional time series modelling problems, with many variant models relying on discrete-time mechanisms such as recurrent neural networks (RNNs). On the other hand, continuous-time methods have recently gained attraction, especially in the context of irregularly-sampled time series, where they can better handle the data than discrete-time methods. One such class are Gaussian process variational autoencoders (GPVAEs), where the VAE prior is set as a Gaussian process (GP). However, a major limitation of GPVAEs is that it inherits the cubic computational cost as GPs, making it unattractive to practioners. In this work, we leverage the equivalent discrete state space representation of Markovian GPs to enable linear time GPVAE training via Kalman filtering and smoothing. We show on a variety of high-dimensional temporal and spatiotemporal tasks that our method performs favourably compared to existing approaches whilst being computationally highly scalable.

X-ray is widely applied for transmission imaging due to its stronger penetration than natural light. When rendering novel view X-ray projections, existing methods mainly based on NeRF suffer from long training time and slow inference speed. In this paper, we propose a 3D Gaussian splatting-based framework, namely X-Gaussian, for X-ray novel view synthesis. Firstly, we redesign a radiative Gaussian point cloud model inspired by the isotropic nature of X-ray imaging. Our model excludes the influence of view direction when learning to predict the radiation intensity of 3D points. Based on this model, we develop a Differentiable Radiative Rasterization (DRR) with CUDA implementation. Secondly, we customize an Angle-pose Cuboid Uniform Initialization (ACUI) strategy that directly uses the parameters of the X-ray scanner to compute the camera information and then uniformly samples point positions within a cuboid enclosing the scanned object. Experiments show that our X-Gaussian outperforms state-of-the-art methods by 6.5 dB while enjoying less than 15% training time and over 73x inference speed. The application on sparse-view CT reconstruction also reveals the practical values of our method. Code and models will be publicly available at https://github.com/caiyuanhao1998/X-Gaussian . A video demo of the training process visualization is at https://www.youtube.com/watch?v=gDVf_Ngeghg .

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.

We introduce GaussianAvatar-Editor, an innovative framework for text-driven editing of animatable Gaussian head avatars that can be fully controlled in expression, pose, and viewpoint. Unlike static 3D Gaussian editing, editing animatable 4D Gaussian avatars presents challenges related to motion occlusion and spatial-temporal inconsistency. To address these issues, we propose the Weighted Alpha Blending Equation (WABE). This function enhances the blending weight of visible Gaussians while suppressing the influence on non-visible Gaussians, effectively handling motion occlusion during editing. Furthermore, to improve editing quality and ensure 4D consistency, we incorporate conditional adversarial learning into the editing process. This strategy helps to refine the edited results and maintain consistency throughout the animation. By integrating these methods, our GaussianAvatar-Editor achieves photorealistic and consistent results in animatable 4D Gaussian editing. We conduct comprehensive experiments across various subjects to validate the effectiveness of our proposed techniques, which demonstrates the superiority of our approach over existing methods. More results and code are available at: [Project Link](https://xiangyueliu.github.io/GaussianAvatar-Editor/).

Clustering is a widely deployed unsupervised learning tool. Model-based clustering is a flexible framework to tackle data heterogeneity when the clusters have different shapes. Likelihood-based inference for mixture distributions often involves non-convex and high-dimensional objective functions, imposing difficult computational and statistical challenges. The classic expectation-maximization (EM) algorithm is a computationally thrifty iterative method that maximizes a surrogate function minorizing the log-likelihood of observed data in each iteration, which however suffers from bad local maxima even in the special case of the standard Gaussian mixture model with common isotropic covariance matrices. On the other hand, recent studies reveal that the unique global solution of a semidefinite programming (SDP) relaxed K-means achieves the information-theoretically sharp threshold for perfectly recovering the cluster labels under the standard Gaussian mixture model. In this paper, we extend the SDP approach to a general setting by integrating cluster labels as model parameters and propose an iterative likelihood adjusted SDP (iLA-SDP) method that directly maximizes the exact observed likelihood in the presence of data heterogeneity. By lifting the cluster assignment to group-specific membership matrices, iLA-SDP avoids centroids estimation -- a key feature that allows exact recovery under well-separateness of centroids without being trapped by their adversarial configurations. Thus iLA-SDP is less sensitive than EM to initialization and more stable on high-dimensional data. Our numeric experiments demonstrate that iLA-SDP can achieve lower mis-clustering errors over several widely used clustering methods including K-means, SDP and EM algorithms.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids clustering. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not exist for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains and applications. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm that achieve an O(k)-fold speedup in the second ("SWAP") phase of the algorithm, but will still find the same results as the original PAM algorithm. If we relax the choice of swaps performed (while retaining comparable quality), we can further accelerate the algorithm by eagerly performing additional swaps in each iteration. With the substantially faster SWAP, we can now explore faster initialization strategies, because (i) the classic ("BUILD") initialization now becomes the bottleneck, and (ii) our swap is fast enough to compensate for worse starting conditions. We also show how the CLARA and CLARANS algorithms benefit from the proposed modifications. While we do not study the parallelization of our approach in this work, it can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100,200, we observed a 458x respectively 1191x speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets, and in particular to higher k.

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of K-sparse degree-d PTFs on R^n, where any such concept depends only on K out of n attributes of the input. Our main contribution is a new algorithm that runs in time ({nd}/{epsilon})^{O(d)} and under the Gaussian marginal distribution, PAC learns the class up to error rate epsilon with O(K^{4d}{epsilon^{2d}} cdot log^{5d} n) samples even when an eta leq O(epsilon^d) fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-2d polynomial as a filter to detect corrupted samples.

Recently, 3D Gaussian splatting (3D-GS) has gained popularity in novel-view scene synthesis. It addresses the challenges of lengthy training times and slow rendering speeds associated with Neural Radiance Fields (NeRFs). Through rapid, differentiable rasterization of 3D Gaussians, 3D-GS achieves real-time rendering and accelerated training. They, however, demand substantial memory resources for both training and storage, as they require millions of Gaussians in their point cloud representation for each scene. We present a technique utilizing quantized embeddings to significantly reduce memory storage requirements and a coarse-to-fine training strategy for a faster and more stable optimization of the Gaussian point clouds. Our approach results in scene representations with fewer Gaussians and quantized representations, leading to faster training times and rendering speeds for real-time rendering of high resolution scenes. We reduce memory by more than an order of magnitude all while maintaining the reconstruction quality. We validate the effectiveness of our approach on a variety of datasets and scenes preserving the visual quality while consuming 10-20x less memory and faster training/inference speed. Project page and code is available https://efficientgaussian.github.io

Novel view synthesis of dynamic scenes has been an intriguing yet challenging problem. Despite recent advancements, simultaneously achieving high-resolution photorealistic results, real-time rendering, and compact storage remains a formidable task. To address these challenges, we propose Spacetime Gaussian Feature Splatting as a novel dynamic scene representation, composed of three pivotal components. First, we formulate expressive Spacetime Gaussians by enhancing 3D Gaussians with temporal opacity and parametric motion/rotation. This enables Spacetime Gaussians to capture static, dynamic, as well as transient content within a scene. Second, we introduce splatted feature rendering, which replaces spherical harmonics with neural features. These features facilitate the modeling of view- and time-dependent appearance while maintaining small size. Third, we leverage the guidance of training error and coarse depth to sample new Gaussians in areas that are challenging to converge with existing pipelines. Experiments on several established real-world datasets demonstrate that our method achieves state-of-the-art rendering quality and speed, while retaining compact storage. At 8K resolution, our lite-version model can render at 60 FPS on an Nvidia RTX 4090 GPU.

Averaging the parameters of models that have the same architecture and initialization can provide a means of combining their respective capabilities. In this paper, we take the perspective that this "merging" operation can be seen as choosing parameters that approximately maximize the joint likelihood of the posteriors of the models' parameters. Computing a simple average of the models' parameters therefore corresponds to making an isotropic Gaussian approximation to their posteriors. We develop an alternative merging procedure based on the Laplace approximation where we approximate each model's posterior as a Gaussian distribution whose precision matrix corresponds to its Fisher information. We first show that our "Fisher merging" technique provides a performance boost in settings where simple parameter averaging is currently used -- specifically, robust fine-tuning and model ensembling. Then, we compare merging to standard gradient-based transfer learning and demonstrate that merging enables a fundamentally different method for transferring capabilities across models. Specifically, we show that Fisher merging is competitive with gradient-based transfer learning approaches (while being significantly cheaper) in intermediate-task training and domain-adaptive pre-training. We also show that our merging procedure makes it possible to combine models in previously unexplored ways. We release our code to facilitate future research into methods for merging models.

Gaussian splatting, renowned for its exceptional rendering quality and efficiency, has emerged as a prominent technique in 3D scene representation. However, the substantial data volume of Gaussian splatting impedes its practical utility in real-world applications. Herein, we propose an efficient 3D scene representation, named Compressed Gaussian Splatting (CompGS), which harnesses compact Gaussian primitives for faithful 3D scene modeling with a remarkably reduced data size. To ensure the compactness of Gaussian primitives, we devise a hybrid primitive structure that captures predictive relationships between each other. Then, we exploit a small set of anchor primitives for prediction, allowing the majority of primitives to be encapsulated into highly compact residual forms. Moreover, we develop a rate-constrained optimization scheme to eliminate redundancies within such hybrid primitives, steering our CompGS towards an optimal trade-off between bitrate consumption and representation efficacy. Experimental results show that the proposed CompGS significantly outperforms existing methods, achieving superior compactness in 3D scene representation without compromising model accuracy and rendering quality. Our code will be released on GitHub for further research.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

The quality of many modern machine learning models improves as model complexity increases, an effect that has been quantified, for predictive performance, with the non-monotonic double descent learning curve. Here, we address the overarching question: is there an analogous theory of double descent for models which estimate uncertainty? We provide a partially affirmative and partially negative answer in the setting of Gaussian processes (GP). Under standard assumptions, we prove that higher model quality for optimally-tuned GPs (including uncertainty prediction) under marginal likelihood is realized for larger input dimensions, and therefore exhibits a monotone error curve. After showing that marginal likelihood does not naturally exhibit double descent in the input dimension, we highlight related forms of posterior predictive loss that do exhibit non-monotonicity. Finally, we verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.

Bayesian optimization (BO) has established itself as a leading strategy for efficiently optimizing expensive-to-evaluate functions. Existing BO methods mostly rely on Gaussian process (GP) surrogate models and are not applicable to (doubly-stochastic) Gaussian Cox processes, where the observation process is modulated by a latent intensity function modeled as a GP. In this paper, we propose a novel maximum a posteriori inference of Gaussian Cox processes. It leverages the Laplace approximation and change of kernel technique to transform the problem into a new reproducing kernel Hilbert space, where it becomes more tractable computationally. It enables us to obtain both a functional posterior of the latent intensity function and the covariance of the posterior, thus extending existing works that often focus on specific link functions or estimating the posterior mean. Using the result, we propose a BO framework based on the Gaussian Cox process model and further develop a Nystr\"om approximation for efficient computation. Extensive evaluations on various synthetic and real-world datasets demonstrate significant improvement over state-of-the-art inference solutions for Gaussian Cox processes, as well as effective BO with a wide range of acquisition functions designed through the underlying Gaussian Cox process model.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

We present, GauHuman, a 3D human model with Gaussian Splatting for both fast training (1 ~ 2 minutes) and real-time rendering (up to 189 FPS), compared with existing NeRF-based implicit representation modelling frameworks demanding hours of training and seconds of rendering per frame. Specifically, GauHuman encodes Gaussian Splatting in the canonical space and transforms 3D Gaussians from canonical space to posed space with linear blend skinning (LBS), in which effective pose and LBS refinement modules are designed to learn fine details of 3D humans under negligible computational cost. Moreover, to enable fast optimization of GauHuman, we initialize and prune 3D Gaussians with 3D human prior, while splitting/cloning via KL divergence guidance, along with a novel merge operation for further speeding up. Extensive experiments on ZJU_Mocap and MonoCap datasets demonstrate that GauHuman achieves state-of-the-art performance quantitatively and qualitatively with fast training and real-time rendering speed. Notably, without sacrificing rendering quality, GauHuman can fast model the 3D human performer with ~13k 3D Gaussians.

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices lauritzen2019maximum,slawski2015estimation and even one observation for diagonally dominant M-matrices truell2021maximum. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted ell_1-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

Text-guided diffusion models have revolutionized image and video generation and have also been successfully used for optimization-based 3D object synthesis. Here, we instead focus on the underexplored text-to-4D setting and synthesize dynamic, animated 3D objects using score distillation methods with an additional temporal dimension. Compared to previous work, we pursue a novel compositional generation-based approach, and combine text-to-image, text-to-video, and 3D-aware multiview diffusion models to provide feedback during 4D object optimization, thereby simultaneously enforcing temporal consistency, high-quality visual appearance and realistic geometry. Our method, called Align Your Gaussians (AYG), leverages dynamic 3D Gaussian Splatting with deformation fields as 4D representation. Crucial to AYG is a novel method to regularize the distribution of the moving 3D Gaussians and thereby stabilize the optimization and induce motion. We also propose a motion amplification mechanism as well as a new autoregressive synthesis scheme to generate and combine multiple 4D sequences for longer generation. These techniques allow us to synthesize vivid dynamic scenes, outperform previous work qualitatively and quantitatively and achieve state-of-the-art text-to-4D performance. Due to the Gaussian 4D representation, different 4D animations can be seamlessly combined, as we demonstrate. AYG opens up promising avenues for animation, simulation and digital content creation as well as synthetic data generation.

3D Gaussian Splatting has recently been embraced as a versatile and effective method for scene reconstruction and novel view synthesis, owing to its high-quality results and compatibility with hardware rasterization. Despite its advantages, Gaussian Splatting's reliance on high-quality point cloud initialization by Structure-from-Motion (SFM) algorithms is a significant limitation to be overcome. To this end, we investigate various initialization strategies for Gaussian Splatting and delve into how volumetric reconstructions from Neural Radiance Fields (NeRF) can be utilized to bypass the dependency on SFM data. Our findings demonstrate that random initialization can perform much better if carefully designed and that by employing a combination of improved initialization strategies and structure distillation from low-cost NeRF models, it is possible to achieve equivalent results, or at times even superior, to those obtained from SFM initialization.

Gaussian splatting has achieved impressive improvements for both novel-view synthesis and surface reconstruction from multi-view images. However, current methods still struggle to reconstruct high-quality surfaces from only sparse view input images using Gaussian splatting. In this paper, we propose a novel method called SolidGS to address this problem. We observed that the reconstructed geometry can be severely inconsistent across multi-views, due to the property of Gaussian function in geometry rendering. This motivates us to consolidate all Gaussians by adopting a more solid kernel function, which effectively improves the surface reconstruction quality. With the additional help of geometrical regularization and monocular normal estimation, our method achieves superior performance on the sparse view surface reconstruction than all the Gaussian splatting methods and neural field methods on the widely used DTU, Tanks-and-Temples, and LLFF datasets.

The Gaussian splatting for radiance field rendering method has recently emerged as an efficient approach for accurate scene representation. It optimizes the location, size, color, and shape of a cloud of 3D Gaussian elements to visually match, after projection, or splatting, a set of given images taken from various viewing directions. And yet, despite the proximity of Gaussian elements to the shape boundaries, direct surface reconstruction of objects in the scene is a challenge. We propose a novel approach for surface reconstruction from Gaussian splatting models. Rather than relying on the Gaussian elements' locations as a prior for surface reconstruction, we leverage the superior novel-view synthesis capabilities of 3DGS. To that end, we use the Gaussian splatting model to render pairs of stereo-calibrated novel views from which we extract depth profiles using a stereo matching method. We then combine the extracted RGB-D images into a geometrically consistent surface. The resulting reconstruction is more accurate and shows finer details when compared to other methods for surface reconstruction from Gaussian splatting models, while requiring significantly less compute time compared to other surface reconstruction methods. We performed extensive testing of the proposed method on in-the-wild scenes, taken by a smartphone, showcasing its superior reconstruction abilities. Additionally, we tested the proposed method on the Tanks and Temples benchmark, and it has surpassed the current leading method for surface reconstruction from Gaussian splatting models. Project page: https://gs2mesh.github.io/.

3D Gaussian Splatting (3DGS) has become the de facto method of 3D representation in many vision tasks. This calls for the 3D understanding directly in this representation space. To facilitate the research in this direction, we first build a large-scale dataset of 3DGS using the commonly used ShapeNet and ModelNet datasets. Our dataset ShapeSplat consists of 65K objects from 87 unique categories, whose labels are in accordance with the respective datasets. The creation of this dataset utilized the compute equivalent of 2 GPU years on a TITAN XP GPU. We utilize our dataset for unsupervised pretraining and supervised finetuning for classification and segmentation tasks. To this end, we introduce \textit{Gaussian-MAE}, which highlights the unique benefits of representation learning from Gaussian parameters. Through exhaustive experiments, we provide several valuable insights. In particular, we show that (1) the distribution of the optimized GS centroids significantly differs from the uniformly sampled point cloud (used for initialization) counterpart; (2) this change in distribution results in degradation in classification but improvement in segmentation tasks when using only the centroids; (3) to leverage additional Gaussian parameters, we propose Gaussian feature grouping in a normalized feature space, along with splats pooling layer, offering a tailored solution to effectively group and embed similar Gaussians, which leads to notable improvement in finetuning tasks.

A key property of deep neural networks (DNNs) is their ability to learn new features during training. This intriguing aspect of deep learning stands out most clearly in recently reported Grokking phenomena. While mainly reflected as a sudden increase in test accuracy, Grokking is also believed to be a beyond lazy-learning/Gaussian Process (GP) phenomenon involving feature learning. Here we apply a recent development in the theory of feature learning, the adaptive kernel approach, to two teacher-student models with cubic-polynomial and modular addition teachers. We provide analytical predictions on feature learning and Grokking properties of these models and demonstrate a mapping between Grokking and the theory of phase transitions. We show that after Grokking, the state of the DNN is analogous to the mixed phase following a first-order phase transition. In this mixed phase, the DNN generates useful internal representations of the teacher that are sharply distinct from those before the transition.

This work presents a fast and scalable algorithm for incremental learning of Gaussian mixture models. By performing rank-one updates on its precision matrices and determinants, its asymptotic time complexity is of NKD^2 for N data points, K Gaussian components and D dimensions. The resulting algorithm can be applied to high dimensional tasks, and this is confirmed by applying it to the classification datasets MNIST and CIFAR-10. Additionally, in order to show the algorithm's applicability to function approximation and control tasks, it is applied to three reinforcement learning tasks and its data-efficiency is evaluated.

Modeling, understanding, and reconstructing the real world are crucial in XR/VR. Recently, 3D Gaussian Splatting (3D-GS) methods have shown remarkable success in modeling and understanding 3D scenes. Similarly, various 4D representations have demonstrated the ability to capture the dynamics of the 4D world. However, there is a dearth of research focusing on segmentation within 4D representations. In this paper, we propose Segment Any 4D Gaussians (SA4D), one of the first frameworks to segment anything in the 4D digital world based on 4D Gaussians. In SA4D, an efficient temporal identity feature field is introduced to handle Gaussian drifting, with the potential to learn precise identity features from noisy and sparse input. Additionally, a 4D segmentation refinement process is proposed to remove artifacts. Our SA4D achieves precise, high-quality segmentation within seconds in 4D Gaussians and shows the ability to remove, recolor, compose, and render high-quality anything masks. More demos are available at: https://jsxzs.github.io/sa4d/.

Recent studies in Radiance Fields have paved the robust way for novel view synthesis with their photorealistic rendering quality. Nevertheless, they usually employ neural networks and volumetric rendering, which are costly to train and impede their broad use in various real-time applications due to the lengthy rendering time. Lately 3D Gaussians splatting-based approach has been proposed to model the 3D scene, and it achieves remarkable visual quality while rendering the images in real-time. However, it suffers from severe degradation in the rendering quality if the training images are blurry. Blurriness commonly occurs due to the lens defocusing, object motion, and camera shake, and it inevitably intervenes in clean image acquisition. Several previous studies have attempted to render clean and sharp images from blurry input images using neural fields. The majority of those works, however, are designed only for volumetric rendering-based neural radiance fields and are not straightforwardly applicable to rasterization-based 3D Gaussian splatting methods. Thus, we propose a novel real-time deblurring framework, deblurring 3D Gaussian Splatting, using a small Multi-Layer Perceptron (MLP) that manipulates the covariance of each 3D Gaussian to model the scene blurriness. While deblurring 3D Gaussian Splatting can still enjoy real-time rendering, it can reconstruct fine and sharp details from blurry images. A variety of experiments have been conducted on the benchmark, and the results have revealed the effectiveness of our approach for deblurring. Qualitative results are available at https://benhenryl.github.io/Deblurring-3D-Gaussian-Splatting/

This is only a preview version of GauMesh. Recently, primitive-based rendering has been proven to achieve convincing results in solving the problem of modeling and rendering the 3D dynamic scene from 2D images. Despite this, in the context of novel view synthesis, each type of primitive has its inherent defects in terms of representation ability. It is difficult to exploit the mesh to depict the fuzzy geometry. Meanwhile, the point-based splatting (e.g. the 3D Gaussian Splatting) method usually produces artifacts or blurry pixels in the area with smooth geometry and sharp textures. As a result, it is difficult, even not impossible, to represent the complex and dynamic scene with a single type of primitive. To this end, we propose a novel approach, GauMesh, to bridge the 3D Gaussian and Mesh for modeling and rendering the dynamic scenes. Given a sequence of tracked mesh as initialization, our goal is to simultaneously optimize the mesh geometry, color texture, opacity maps, a set of 3D Gaussians, and the deformation field. At a specific time, we perform alpha-blending on the RGB and opacity values based on the merged and re-ordered z-buffers from mesh and 3D Gaussian rasterizations. This produces the final rendering, which is supervised by the ground-truth image. Experiments demonstrate that our approach adapts the appropriate type of primitives to represent the different parts of the dynamic scene and outperforms all the baseline methods in both quantitative and qualitative comparisons without losing render speed.

Creating 4D fields of Gaussian Splatting from images or videos is a challenging task due to its under-constrained nature. While the optimization can draw photometric reference from the input videos or be regulated by generative models, directly supervising Gaussian motions remains underexplored. In this paper, we introduce a novel concept, Gaussian flow, which connects the dynamics of 3D Gaussians and pixel velocities between consecutive frames. The Gaussian flow can be efficiently obtained by splatting Gaussian dynamics into the image space. This differentiable process enables direct dynamic supervision from optical flow. Our method significantly benefits 4D dynamic content generation and 4D novel view synthesis with Gaussian Splatting, especially for contents with rich motions that are hard to be handled by existing methods. The common color drifting issue that happens in 4D generation is also resolved with improved Guassian dynamics. Superior visual quality on extensive experiments demonstrates our method's effectiveness. Quantitative and qualitative evaluations show that our method achieves state-of-the-art results on both tasks of 4D generation and 4D novel view synthesis. Project page: https://zerg-overmind.github.io/GaussianFlow.github.io/

Dense simultaneous localization and mapping (SLAM) is pivotal for embodied scene understanding. Recent work has shown that 3D Gaussians enable high-quality reconstruction and real-time rendering of scenes using multiple posed cameras. In this light, we show for the first time that representing a scene by 3D Gaussians can enable dense SLAM using a single unposed monocular RGB-D camera. Our method, SplaTAM, addresses the limitations of prior radiance field-based representations, including fast rendering and optimization, the ability to determine if areas have been previously mapped, and structured map expansion by adding more Gaussians. We employ an online tracking and mapping pipeline while tailoring it to specifically use an underlying Gaussian representation and silhouette-guided optimization via differentiable rendering. Extensive experiments show that SplaTAM achieves up to 2X state-of-the-art performance in camera pose estimation, map construction, and novel-view synthesis, demonstrating its superiority over existing approaches, while allowing real-time rendering of a high-resolution dense 3D map.

Reversing a diffusion process by learning its score forms the heart of diffusion-based generative modeling and for estimating properties of scientific systems. The diffusion processes that are tractable center on linear processes with a Gaussian stationary distribution. This limits the kinds of models that can be built to those that target a Gaussian prior or more generally limits the kinds of problems that can be generically solved to those that have conditionally linear score functions. In this work, we introduce a family of tractable denoising score matching objectives, called local-DSM, built using local increments of the diffusion process. We show how local-DSM melded with Taylor expansions enables automated training and score estimation with nonlinear diffusion processes. To demonstrate these ideas, we use automated-DSM to train generative models using non-Gaussian priors on challenging low dimensional distributions and the CIFAR10 image dataset. Additionally, we use the automated-DSM to learn the scores for nonlinear processes studied in statistical physics.

Neural Radiance Fields (NeRFs) have demonstrated remarkable potential in capturing complex 3D scenes with high fidelity. However, one persistent challenge that hinders the widespread adoption of NeRFs is the computational bottleneck due to the volumetric rendering. On the other hand, 3D Gaussian splatting (3DGS) has recently emerged as an alternative representation that leverages a 3D Gaussisan-based representation and adopts the rasterization pipeline to render the images rather than volumetric rendering, achieving very fast rendering speed and promising image quality. However, a significant drawback arises as 3DGS entails a substantial number of 3D Gaussians to maintain the high fidelity of the rendered images, which requires a large amount of memory and storage. To address this critical issue, we place a specific emphasis on two key objectives: reducing the number of Gaussian points without sacrificing performance and compressing the Gaussian attributes, such as view-dependent color and covariance. To this end, we propose a learnable mask strategy that significantly reduces the number of Gaussians while preserving high performance. In addition, we propose a compact but effective representation of view-dependent color by employing a grid-based neural field rather than relying on spherical harmonics. Finally, we learn codebooks to compactly represent the geometric attributes of Gaussian by vector quantization. In our extensive experiments, we consistently show over 10times reduced storage and enhanced rendering speed, while maintaining the quality of the scene representation, compared to 3DGS. Our work provides a comprehensive framework for 3D scene representation, achieving high performance, fast training, compactness, and real-time rendering. Our project page is available at https://maincold2.github.io/c3dgs/.

Deep generative models are routinely used in generating samples from complex, high-dimensional distributions. Despite their apparent successes, their statistical properties are not well understood. A common assumption is that with enough training data and sufficiently large neural networks, deep generative model samples will have arbitrarily small errors in sampling from any continuous target distribution. We set up a unifying framework that debunks this belief. We demonstrate that broad classes of deep generative models, including variational autoencoders and generative adversarial networks, are not universal generators. Under the predominant case of Gaussian latent variables, these models can only generate concentrated samples that exhibit light tails. Using tools from concentration of measure and convex geometry, we give analogous results for more general log-concave and strongly log-concave latent variable distributions. We extend our results to diffusion models via a reduction argument. We use the Gromov--Levy inequality to give similar guarantees when the latent variables lie on manifolds with positive Ricci curvature. These results shed light on the limited capacity of common deep generative models to handle heavy tails. We illustrate the empirical relevance of our work with simulations and financial data.

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

3D Gaussian Splatting (3DGS) is increasingly popular for 3D reconstruction due to its superior visual quality and rendering speed. However, 3DGS training currently occurs on a single GPU, limiting its ability to handle high-resolution and large-scale 3D reconstruction tasks due to memory constraints. We introduce Grendel, a distributed system designed to partition 3DGS parameters and parallelize computation across multiple GPUs. As each Gaussian affects a small, dynamic subset of rendered pixels, Grendel employs sparse all-to-all communication to transfer the necessary Gaussians to pixel partitions and performs dynamic load balancing. Unlike existing 3DGS systems that train using one camera view image at a time, Grendel supports batched training with multiple views. We explore various optimization hyperparameter scaling strategies and find that a simple sqrt(batch size) scaling rule is highly effective. Evaluations using large-scale, high-resolution scenes show that Grendel enhances rendering quality by scaling up 3DGS parameters across multiple GPUs. On the Rubble dataset, we achieve a test PSNR of 27.28 by distributing 40.4 million Gaussians across 16 GPUs, compared to a PSNR of 26.28 using 11.2 million Gaussians on a single GPU. Grendel is an open-source project available at: https://github.com/nyu-systems/Grendel-GS

We propose MVSplat, an efficient feed-forward 3D Gaussian Splatting model learned from sparse multi-view images. To accurately localize the Gaussian centers, we propose to build a cost volume representation via plane sweeping in the 3D space, where the cross-view feature similarities stored in the cost volume can provide valuable geometry cues to the estimation of depth. We learn the Gaussian primitives' opacities, covariances, and spherical harmonics coefficients jointly with the Gaussian centers while only relying on photometric supervision. We demonstrate the importance of the cost volume representation in learning feed-forward Gaussian Splatting models via extensive experimental evaluations. On the large-scale RealEstate10K and ACID benchmarks, our model achieves state-of-the-art performance with the fastest feed-forward inference speed (22 fps). Compared to the latest state-of-the-art method pixelSplat, our model uses 10times fewer parameters and infers more than 2times faster while providing higher appearance and geometry quality as well as better cross-dataset generalization.

State-of-the-art novel view synthesis methods achieve impressive results for multi-view captures of static 3D scenes. However, the reconstructed scenes still lack "liveliness," a key component for creating engaging 3D experiences. Recently, novel video diffusion models generate realistic videos with complex motion and enable animations of 2D images, however they cannot naively be used to animate 3D scenes as they lack multi-view consistency. To breathe life into the static world, we propose Gaussians2Life, a method for animating parts of high-quality 3D scenes in a Gaussian Splatting representation. Our key idea is to leverage powerful video diffusion models as the generative component of our model and to combine these with a robust technique to lift 2D videos into meaningful 3D motion. We find that, in contrast to prior work, this enables realistic animations of complex, pre-existing 3D scenes and further enables the animation of a large variety of object classes, while related work is mostly focused on prior-based character animation, or single 3D objects. Our model enables the creation of consistent, immersive 3D experiences for arbitrary scenes.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not hold for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains such as biology that require the use of Jaccard, Gower, or more complex distances. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm to achieve an O(k)-fold speedup in the second SWAP phase of the algorithm, but will still find the same results as the original PAM algorithm. If we slightly relax the choice of swaps performed (at comparable quality), we can further accelerate the algorithm by performing up to k swaps in each iteration. With the substantially faster SWAP, we can now also explore alternative strategies for choosing the initial medoids. We also show how the CLARA and CLARANS algorithms benefit from these modifications. It can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100, we observed a 200-fold speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets as long as we can afford to compute a distance matrix, and in particular to higher k (at k=2, the new SWAP was only 1.5 times faster, as the speedup is expected to increase with k).

Recent advancements in 3D reconstruction from single images have been driven by the evolution of generative models. Prominent among these are methods based on Score Distillation Sampling (SDS) and the adaptation of diffusion models in the 3D domain. Despite their progress, these techniques often face limitations due to slow optimization or rendering processes, leading to extensive training and optimization times. In this paper, we introduce a novel approach for single-view reconstruction that efficiently generates a 3D model from a single image via feed-forward inference. Our method utilizes two transformer-based networks, namely a point decoder and a triplane decoder, to reconstruct 3D objects using a hybrid Triplane-Gaussian intermediate representation. This hybrid representation strikes a balance, achieving a faster rendering speed compared to implicit representations while simultaneously delivering superior rendering quality than explicit representations. The point decoder is designed for generating point clouds from single images, offering an explicit representation which is then utilized by the triplane decoder to query Gaussian features for each point. This design choice addresses the challenges associated with directly regressing explicit 3D Gaussian attributes characterized by their non-structural nature. Subsequently, the 3D Gaussians are decoded by an MLP to enable rapid rendering through splatting. Both decoders are built upon a scalable, transformer-based architecture and have been efficiently trained on large-scale 3D datasets. The evaluations conducted on both synthetic datasets and real-world images demonstrate that our method not only achieves higher quality but also ensures a faster runtime in comparison to previous state-of-the-art techniques. Please see our project page at https://zouzx.github.io/TriplaneGaussian/.

Gaussian process state-space models (GPSSMs) provide a principled and flexible approach to modeling the dynamics of a latent state, which is observed at discrete-time points via a likelihood model. However, inference in GPSSMs is computationally and statistically challenging due to the large number of latent variables in the model and the strong temporal dependencies between them. In this paper, we propose a new method for inference in Bayesian GPSSMs, which overcomes the drawbacks of previous approaches, namely over-simplified assumptions, and high computational requirements. Our method is based on free-form variational inference via stochastic gradient Hamiltonian Monte Carlo within the inducing-variable formalism. Furthermore, by exploiting our proposed variational distribution, we provide a collapsed extension of our method where the inducing variables are marginalized analytically. We also showcase results when combining our framework with particle MCMC methods. We show that, on six real-world datasets, our approach can learn transition dynamics and latent states more accurately than competing methods.

In novel view synthesis of scenes from multiple input views, 3D Gaussian splatting emerges as a viable alternative to existing radiance field approaches, delivering great visual quality and real-time rendering. While successful in static scenes, the present advancement of 3D Gaussian representation, however, faces challenges in dynamic scenes in terms of memory consumption and the need for numerous observations per time step, due to the onus of storing 3D Gaussian parameters per time step. In this study, we present an efficient 3D Gaussian representation tailored for dynamic scenes in which we define positions and rotations as functions of time while leaving other time-invariant properties of the static 3D Gaussian unchanged. Notably, our representation reduces memory usage, which is consistent regardless of the input sequence length. Additionally, it mitigates the risk of overfitting observed frames by accounting for temporal changes. The optimization of our Gaussian representation based on image and flow reconstruction results in a powerful framework for dynamic scene view synthesis in both monocular and multi-view cases. We obtain the highest rendering speed of 118 frames per second (FPS) at a resolution of 1352 times 1014 with a single GPU, showing the practical usability and effectiveness of our proposed method in dynamic scene rendering scenarios.

Reconstructing realistic 3D human models from monocular images has significant applications in creative industries, human-computer interfaces, and healthcare. We base our work on 3D Gaussian Splatting (3DGS), a scene representation composed of a mixture of Gaussians. Predicting such mixtures for a human from a single input image is challenging, as it is a non-uniform density (with a many-to-one relationship with input pixels) with strict physical constraints. At the same time, it needs to be flexible to accommodate a variety of clothes and poses. Our key observation is that the vertices of standardized human meshes (such as SMPL) can provide an adequate density and approximate initial position for Gaussians. We can then train a transformer model to jointly predict comparatively small adjustments to these positions, as well as the other Gaussians' attributes and the SMPL parameters. We show empirically that this combination (using only multi-view supervision) can achieve fast inference of 3D human models from a single image without test-time optimization, expensive diffusion models, or 3D points supervision. We also show that it can improve 3D pose estimation by better fitting human models that account for clothes and other variations. The code is available on the project website https://abdullahamdi.com/gst/ .

The sliced Wasserstein (SW) distances between two probability measures are defined as the expectation of the Wasserstein distance between two one-dimensional projections of the two measures. The randomness comes from a projecting direction that is used to project the two input measures to one dimension. Due to the intractability of the expectation, Monte Carlo integration is performed to estimate the value of the SW distance. Despite having various variants, there has been no prior work that improves the Monte Carlo estimation scheme for the SW distance in terms of controlling its variance. To bridge the literature on variance reduction and the literature on the SW distance, we propose computationally efficient control variates to reduce the variance of the empirical estimation of the SW distance. The key idea is to first find Gaussian approximations of projected one-dimensional measures, then we utilize the closed-form of the Wasserstein-2 distance between two Gaussian distributions to design the control variates. In particular, we propose using a lower bound and an upper bound of the Wasserstein-2 distance between two fitted Gaussians as two computationally efficient control variates. We empirically show that the proposed control variate estimators can help to reduce the variance considerably when comparing measures over images and point-clouds. Finally, we demonstrate the favorable performance of the proposed control variate estimators in gradient flows to interpolate between two point-clouds and in deep generative modeling on standard image datasets, such as CIFAR10 and CelebA.

We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte Carlo (MCMC) methods derived from discretizations of the overdamped Langevin diffusions, which are commonly known as Langevin Monte Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases for which a large class of proximal MCMC methods have been developed: (i) a nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling tasks arise from a wide range of applications to Bayesian inference and imaging inverse problems such as image deconvolution. We perform numerical simulations to compare the performance of most commonly used Langevin Monte Carlo algorithms.

Achieving high-resolution novel view synthesis (HRNVS) from low-resolution input views is a challenging task due to the lack of high-resolution data. Previous methods optimize high-resolution Neural Radiance Field (NeRF) from low-resolution input views but suffer from slow rendering speed. In this work, we base our method on 3D Gaussian Splatting (3DGS) due to its capability of producing high-quality images at a faster rendering speed. To alleviate the shortage of data for higher-resolution synthesis, we propose to leverage off-the-shelf 2D diffusion priors by distilling the 2D knowledge into 3D with Score Distillation Sampling (SDS). Nevertheless, applying SDS directly to Gaussian-based 3D super-resolution leads to undesirable and redundant 3D Gaussian primitives, due to the randomness brought by generative priors. To mitigate this issue, we introduce two simple yet effective techniques to reduce stochastic disturbances introduced by SDS. Specifically, we 1) shrink the range of diffusion timestep in SDS with an annealing strategy; 2) randomly discard redundant Gaussian primitives during densification. Extensive experiments have demonstrated that our proposed GaussainSR can attain high-quality results for HRNVS with only low-resolution inputs on both synthetic and real-world datasets. Project page: https://chchnii.github.io/GaussianSR/

This paper investigates the problem of generalized linear bandits with heavy-tailed rewards, whose (1+epsilon)-th moment is bounded for some epsilonin (0,1]. Although there exist methods for generalized linear bandits, most of them focus on bounded or sub-Gaussian rewards and are not well-suited for many real-world scenarios, such as financial markets and web-advertising. To address this issue, we propose two novel algorithms based on truncation and mean of medians. These algorithms achieve an almost optimal regret bound of O(dT^{1{1+epsilon}}), where d is the dimension of contextual information and T is the time horizon. Our truncation-based algorithm supports online learning, distinguishing it from existing truncation-based approaches. Additionally, our mean-of-medians-based algorithm requires only O(log T) rewards and one estimator per epoch, making it more practical. Moreover, our algorithms improve the regret bounds by a logarithmic factor compared to existing algorithms when epsilon=1. Numerical experimental results confirm the merits of our algorithms.

This work builds upon previous efforts in online incremental learning, namely the Incremental Gaussian Mixture Network (IGMN). The IGMN is capable of learning from data streams in a single-pass by improving its model after analyzing each data point and discarding it thereafter. Nevertheless, it suffers from the scalability point-of-view, due to its asymptotic time complexity of Obigl(NKD^3bigr) for N data points, K Gaussian components and D dimensions, rendering it inadequate for high-dimensional data. In this paper, we manage to reduce this complexity to Obigl(NKD^2bigr) by deriving formulas for working directly with precision matrices instead of covariance matrices. The final result is a much faster and scalable algorithm which can be applied to high dimensional tasks. This is confirmed by applying the modified algorithm to high-dimensional classification datasets.

Recently, image-to-3D approaches have significantly advanced the generation quality and speed of 3D assets based on large reconstruction models, particularly 3D Gaussian reconstruction models. Existing large 3D Gaussian models directly map 2D image to 3D Gaussian parameters, while regressing 2D image to 3D Gaussian representations is challenging without 3D priors. In this paper, we propose a large Point-to-Gaussian model, that inputs the initial point cloud produced from large 3D diffusion model conditional on 2D image to generate the Gaussian parameters, for image-to-3D generation. The point cloud provides initial 3D geometry prior for Gaussian generation, thus significantly facilitating image-to-3D Generation. Moreover, we present the Attention mechanism, Projection mechanism, and Point feature extractor, dubbed as APP block, for fusing the image features with point cloud features. The qualitative and quantitative experiments extensively demonstrate the effectiveness of the proposed approach on GSO and Objaverse datasets, and show the proposed method achieves state-of-the-art performance.

We introduce Gaussian-Flow, a novel point-based approach for fast dynamic scene reconstruction and real-time rendering from both multi-view and monocular videos. In contrast to the prevalent NeRF-based approaches hampered by slow training and rendering speeds, our approach harnesses recent advancements in point-based 3D Gaussian Splatting (3DGS). Specifically, a novel Dual-Domain Deformation Model (DDDM) is proposed to explicitly model attribute deformations of each Gaussian point, where the time-dependent residual of each attribute is captured by a polynomial fitting in the time domain, and a Fourier series fitting in the frequency domain. The proposed DDDM is capable of modeling complex scene deformations across long video footage, eliminating the need for training separate 3DGS for each frame or introducing an additional implicit neural field to model 3D dynamics. Moreover, the explicit deformation modeling for discretized Gaussian points ensures ultra-fast training and rendering of a 4D scene, which is comparable to the original 3DGS designed for static 3D reconstruction. Our proposed approach showcases a substantial efficiency improvement, achieving a 5times faster training speed compared to the per-frame 3DGS modeling. In addition, quantitative results demonstrate that the proposed Gaussian-Flow significantly outperforms previous leading methods in novel view rendering quality. Project page: https://nju-3dv.github.io/projects/Gaussian-Flow

We present a web-based software tool, the Virtual Quantum Optics Laboratory (VQOL), that may be used for designing and executing realistic simulations of quantum optics experiments. A graphical user interface allows one to rapidly build and configure a variety of different optical experiments, while the runtime environment provides unique capabilities for visualization and analysis. All standard linear optical components are available as well as sources of thermal, coherent, and entangled Gaussian states. A unique aspect of VQOL is the introduction of non-Gaussian measurements using detectors modeled as deterministic devices that "click" when the amplitude of the light falls above a given threshold. We describe the underlying theoretical models and provide several illustrative examples. We find that VQOL provides a a faithful representation of many experimental quantum optics phenomena and may serve as both a useful instructional tool for students as well as a valuable research tool for practitioners.

3D Gaussian Splatting (GS) have achieved considerable improvement over Neural Radiance Fields in terms of 3D fitting fidelity and rendering speed. However, this unstructured representation with scattered Gaussians poses a significant challenge for generative modeling. To address the problem, we introduce GaussianCube, a structured GS representation that is both powerful and efficient for generative modeling. We achieve this by first proposing a modified densification-constrained GS fitting algorithm which can yield high-quality fitting results using a fixed number of free Gaussians, and then re-arranging the Gaussians into a predefined voxel grid via Optimal Transport. The structured grid representation allows us to use standard 3D U-Net as our backbone in diffusion generative modeling without elaborate designs. Extensive experiments conducted on ShapeNet and OmniObject3D show that our model achieves state-of-the-art generation results both qualitatively and quantitatively, underscoring the potential of GaussianCube as a powerful and versatile 3D representation.

Recent 3D large reconstruction models (LRMs) can generate high-quality 3D content in sub-seconds by integrating multi-view diffusion models with scalable multi-view reconstructors. Current works further leverage 3D Gaussian Splatting as 3D representation for improved visual quality and rendering efficiency. However, we observe that existing Gaussian reconstruction models often suffer from multi-view inconsistency and blurred textures. We attribute this to the compromise of multi-view information propagation in favor of adopting powerful yet computationally intensive architectures (e.g., Transformers). To address this issue, we introduce MVGamba, a general and lightweight Gaussian reconstruction model featuring a multi-view Gaussian reconstructor based on the RNN-like State Space Model (SSM). Our Gaussian reconstructor propagates causal context containing multi-view information for cross-view self-refinement while generating a long sequence of Gaussians for fine-detail modeling with linear complexity. With off-the-shelf multi-view diffusion models integrated, MVGamba unifies 3D generation tasks from a single image, sparse images, or text prompts. Extensive experiments demonstrate that MVGamba outperforms state-of-the-art baselines in all 3D content generation scenarios with approximately only 0.1times of the model size.

Recent approaches representing 3D objects and scenes using Gaussian splats show increased rendering speed across a variety of platforms and devices. While rendering such representations is indeed extremely efficient, storing and transmitting them is often prohibitively expensive. To represent large-scale scenes, one often needs to store millions of 3D Gaussians, occupying gigabytes of disk space. This poses a very practical limitation, prohibiting widespread adoption.Several solutions have been proposed to strike a balance between disk size and rendering quality, noticeably reducing the visual quality. In this work, we propose a new representation that dramatically reduces the hard drive footprint while featuring similar or improved quality when compared to the standard 3D Gaussian splats. When compared to other compact solutions, ours offers higher quality renderings with significantly reduced storage, being able to efficiently run on a mobile device in real-time. Our key observation is that nearby points in the scene can share similar representations. Hence, only a small ratio of 3D points needs to be stored. We introduce an approach to identify such points which are called parent points. The discarded points called children points along with attributes can be efficiently predicted by tiny MLPs.

We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

3D Gaussian Splatting (3DGS) has become an emerging technique with remarkable potential in 3D representation and image rendering. However, the substantial storage overhead of 3DGS significantly impedes its practical applications. In this work, we formulate the compact 3D Gaussian learning as an end-to-end Rate-Distortion Optimization (RDO) problem and propose RDO-Gaussian that can achieve flexible and continuous rate control. RDO-Gaussian addresses two main issues that exist in current schemes: 1) Different from prior endeavors that minimize the rate under the fixed distortion, we introduce dynamic pruning and entropy-constrained vector quantization (ECVQ) that optimize the rate and distortion at the same time. 2) Previous works treat the colors of each Gaussian equally, while we model the colors of different regions and materials with learnable numbers of parameters. We verify our method on both real and synthetic scenes, showcasing that RDO-Gaussian greatly reduces the size of 3D Gaussian over 40x, and surpasses existing methods in rate-distortion performance.

Conventional uncertainty-aware temporal difference (TD) learning methods often rely on simplistic assumptions, typically including a zero-mean Gaussian distribution for TD errors. Such oversimplification can lead to inaccurate error representations and compromised uncertainty estimation. In this paper, we introduce a novel framework for generalized Gaussian error modeling in deep reinforcement learning, applicable to both discrete and continuous control settings. Our framework enhances the flexibility of error distribution modeling by incorporating additional higher-order moment, particularly kurtosis, thereby improving the estimation and mitigation of data-dependent noise, i.e., aleatoric uncertainty. We examine the influence of the shape parameter of the generalized Gaussian distribution (GGD) on aleatoric uncertainty and provide a closed-form expression that demonstrates an inverse relationship between uncertainty and the shape parameter. Additionally, we propose a theoretically grounded weighting scheme to fully leverage the GGD. To address epistemic uncertainty, we enhance the batch inverse variance weighting by incorporating bias reduction and kurtosis considerations, resulting in improved robustness. Extensive experimental evaluations using policy gradient algorithms demonstrate the consistent efficacy of our method, showcasing significant performance improvements.

In computer graphics, there is a need to recover easily modifiable representations of 3D geometry and appearance from image data. We introduce a novel method for this task using 3D Gaussian Splatting, which enables intuitive scene editing through mesh adjustments. Starting with input images and camera poses, we reconstruct the underlying geometry using a neural Signed Distance Field and extract a high-quality mesh. Our model then estimates a set of Gaussians, where each component is flat, and the opacity is conditioned on the recovered neural surface. To facilitate editing, we produce a proxy representation that encodes information about the Gaussians' shape and position. Unlike other methods, our pipeline allows modifications applied to the extracted mesh to be propagated to the proxy representation, from which we recover the updated parameters of the Gaussians. This effectively transfers the mesh edits back to the recovered appearance representation. By leveraging mesh-guided transformations, our approach simplifies 3D scene editing and offers improvements over existing methods in terms of usability and visual fidelity of edits. The complete source code for this project can be accessed at https://github.com/WJakubowska/NeuralSurfacePriors

3D Gaussian Splatting has emerged as an efficient photorealistic novel view synthesis method. However, its reliance on sparse Structure-from-Motion (SfM) point clouds consistently compromises the scene reconstruction quality. To address these limitations, this paper proposes a novel 3D reconstruction framework Gaussian Processes Gaussian Splatting (GP-GS), where a multi-output Gaussian Process model is developed to achieve adaptive and uncertainty-guided densification of sparse SfM point clouds. Specifically, we propose a dynamic sampling and filtering pipeline that adaptively expands the SfM point clouds by leveraging GP-based predictions to infer new candidate points from the input 2D pixels and depth maps. The pipeline utilizes uncertainty estimates to guide the pruning of high-variance predictions, ensuring geometric consistency and enabling the generation of dense point clouds. The densified point clouds provide high-quality initial 3D Gaussians to enhance reconstruction performance. Extensive experiments conducted on synthetic and real-world datasets across various scales validate the effectiveness and practicality of the proposed framework.

Implicit Neural Representations (INRs) employ neural networks to approximate discrete data as continuous functions. In the context of video data, such models can be utilized to transform the coordinates of pixel locations along with frame occurrence times (or indices) into RGB color values. Although INRs facilitate effective compression, they are unsuitable for editing purposes. One potential solution is to use a 3D Gaussian Splatting (3DGS) based model, such as the Video Gaussian Representation (VGR), which is capable of encoding video as a multitude of 3D Gaussians and is applicable for numerous video processing operations, including editing. Nevertheless, in this case, the capacity for modification is constrained to a limited set of basic transformations. To address this issue, we introduce the Video Gaussian Splatting (VeGaS) model, which enables realistic modifications of video data. To construct VeGaS, we propose a novel family of Folded-Gaussian distributions designed to capture nonlinear dynamics in a video stream and model consecutive frames by 2D Gaussians obtained as respective conditional distributions. Our experiments demonstrate that VeGaS outperforms state-of-the-art solutions in frame reconstruction tasks and allows realistic modifications of video data. The code is available at: https://github.com/gmum/VeGaS.

We aim to address sparse-view reconstruction of a 3D scene by leveraging priors from large-scale vision models. While recent advancements such as 3D Gaussian Splatting (3DGS) have demonstrated remarkable successes in 3D reconstruction, these methods typically necessitate hundreds of input images that densely capture the underlying scene, making them time-consuming and impractical for real-world applications. However, sparse-view reconstruction is inherently ill-posed and under-constrained, often resulting in inferior and incomplete outcomes. This is due to issues such as failed initialization, overfitting on input images, and a lack of details. To mitigate these challenges, we introduce LM-Gaussian, a method capable of generating high-quality reconstructions from a limited number of images. Specifically, we propose a robust initialization module that leverages stereo priors to aid in the recovery of camera poses and the reliable point clouds. Additionally, a diffusion-based refinement is iteratively applied to incorporate image diffusion priors into the Gaussian optimization process to preserve intricate scene details. Finally, we utilize video diffusion priors to further enhance the rendered images for realistic visual effects. Overall, our approach significantly reduces the data acquisition requirements compared to previous 3DGS methods. We validate the effectiveness of our framework through experiments on various public datasets, demonstrating its potential for high-quality 360-degree scene reconstruction. Visual results are on our website.

Gaussian splatting has emerged as a powerful 3D representation that harnesses the advantages of both explicit (mesh) and implicit (NeRF) 3D representations. In this paper, we seek to leverage Gaussian splatting to generate realistic animatable avatars from textual descriptions, addressing the limitations (e.g., flexibility and efficiency) imposed by mesh or NeRF-based representations. However, a naive application of Gaussian splatting cannot generate high-quality animatable avatars and suffers from learning instability; it also cannot capture fine avatar geometries and often leads to degenerate body parts. To tackle these problems, we first propose a primitive-based 3D Gaussian representation where Gaussians are defined inside pose-driven primitives to facilitate animation. Second, to stabilize and amortize the learning of millions of Gaussians, we propose to use neural implicit fields to predict the Gaussian attributes (e.g., colors). Finally, to capture fine avatar geometries and extract detailed meshes, we propose a novel SDF-based implicit mesh learning approach for 3D Gaussians that regularizes the underlying geometries and extracts highly detailed textured meshes. Our proposed method, GAvatar, enables the large-scale generation of diverse animatable avatars using only text prompts. GAvatar significantly surpasses existing methods in terms of both appearance and geometry quality, and achieves extremely fast rendering (100 fps) at 1K resolution.

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P,d) where each point belongs to one of the ell different demographics (i.e., P = P_1 uplus P_2 uplus cdots uplus P_ell) and a set of ell intervals [alpha_1, beta_1], cdots, [alpha_ell, beta_ell] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum ell_p-clustering cost (i.e., (sum_{vin P} d(v,C)^p)^{1/p}) such that for each group iin ell, |Ccap P_i| in [alpha_i, beta_i]. In particular, the fair range ell_p-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range ell_p-clustering for all values of pin [1,infty).

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

Recent progress in neural rendering has brought forth pioneering methods, such as NeRF and Gaussian Splatting, which revolutionize view rendering across various domains like AR/VR, gaming, and content creation. While these methods excel at interpolating {\em within the training data}, the challenge of generalizing to new scenes and objects from very sparse views persists. Specifically, modeling 3D humans from sparse views presents formidable hurdles due to the inherent complexity of human geometry, resulting in inaccurate reconstructions of geometry and textures. To tackle this challenge, this paper leverages recent advancements in Gaussian Splatting and introduces a new method to learn generalizable human Gaussians that allows photorealistic and accurate view-rendering of a new human subject from a limited set of sparse views in a feed-forward manner. A pivotal innovation of our approach involves reformulating the learning of 3D Gaussian parameters into a regression process defined on the 2D UV space of a human template, which allows leveraging the strong geometry prior and the advantages of 2D convolutions. In addition, a multi-scaffold is proposed to effectively represent the offset details. Our method outperforms recent methods on both within-dataset generalization as well as cross-dataset generalization settings.

This work presents a novel algorithm that integrates a data-efficient function approximator with reinforcement learning in continuous state spaces. An online and incremental algorithm capable of learning from a single pass through data, called Incremental Gaussian Mixture Network (IGMN), was employed as a sample-efficient function approximator for the joint state and Q-values space, all in a single model, resulting in a concise and data-efficient algorithm, i.e., a reinforcement learning algorithm that learns from very few interactions with the environment. Results are analyzed to explain the properties of the obtained algorithm, and it is observed that the use of the IGMN function approximator brings some important advantages to reinforcement learning in relation to conventional neural networks trained by gradient descent methods.