We believe that the techniques used to overcome missing data problem in other domains can also be employed in software engineering . In this paper we propose Matrix Factorization algorithm to tackle with missing data problem in building predictive models in software development domain . We provide an upper-bound for the bias of the proposed method . Our experiments show that , our method is 40 times as fast as the baseline method on finding the optimal model order in a matrix factorization problem . Results Watson ’s natural language processing generated a relationship network matching 55 ( 86 % ) genes upstream of BTK and 98 ( 95 % ) genes downstream of Bruton ’s tyrosine kinase in a prominent manually curated physical interaction network . Matrix factorization analysis predicted 11 of 13 genes identified by Merck subject matter experts in the top 20 % of Watson for Drug Discovery ’s 13,595 ranked genes , with 7 in the top 5 % . Conclusion Taken together , these results suggest that Watson for Drug Discovery ’s automatic relationship network identifies the majority of upstream and downstream genes in biological pathway networks and can be used to help with the identification and prioritization of pharmacodynamic biomarker evaluation , accelerating the early phases of disease hypothesis generation . This has been recently exploited for the fusion of low spatial resolution HSI with high spatial resolution multispectral images in order to obtain super-resolution HSI . Most approaches adopt an unmixing or a matrix factorization perspective . The derived methods have led to state-of-the-art results when the spectral information lies in a low-dimensional subspace/manifold . This allows us to rank new items for this user . We then enrich the item and user representations in order to perform rating prediction using a classical matrix factorization scheme . We demonstrate the interest of our approach regarding both item ranking and rating prediction on a series of classical benchmarks . Other than known challenges in collaborative filtering techniques , accuracy and computational cost at a large scale data still at saliency . In this paper we proposed an approach by utilizing matrix value factorization for predicting rating i by user j with the sub matrix as k-most similar items specific to user i for all users who rated them all . In an attempt , previously predicted values are used for subsequent predictions and we have investigated the accuracy of neighborhood methods by applying our method on Netflix Prize ( http://www.netflixprize.com/ ) . The disadvantage of the traditional CFAbMD algorithm is no consideration of impact of local users ’ neighbor on item rating . Aiming at this problem , a new CFAbMD algorithm is proposed considering both ALS matrix factorization and user nearest neighbor ( CFAbMD-UNN ) , which integrates the similarity information among users into the matrix factorization of model . Furthermore , the CFAbMD-UNN algorithm was implemented in parallel on Spark . A digital camera with zoom lenses is used to capture images of the investigated object . The image sequence is then transferred to a Matlab ® code to detect the interest points , then to compute the 3-D coordinates of those interest points and the camera poses by matrix factorization operation . Neither calibration object nor a priori information of the camera is needed . Moreover , users are usually more interested in the top 20 or even top 10 recommended POIs , which makes personalized ranking important in this task . From previous work , directly optimizing for pairwise ranking like Bayesian Personalized Ranking ( BPR ) achieves better performance in the top-k recommendation than directly using matrix matrix factorization that aims to minimize the point-wise rating error . To consider users ’ preferences , geographical influence and personalized ranking , we propose a unified POI recommendation framework , which unifies all of them together . The updated hidden state vector is then used to predict a probability distribution over the next sequence element , using function . In the case where consists of mutually exclusive discrete outcomes , may apply a matrix multiplication followed by a softmax function : Generative RNNs can evaluate log - likelihoods of sequences exactly , and are differentiable with respect to these log - likelihoods . RNNs can be difficult to train due to the vanishing gradient problem , but advances such as the long short - term memory architecture ( LSTM ) have allowed RNNs to be successful . Spectral clustering : This is a matrix factorization approach in which we take the top eigenvectors of the normalized Laplacian matrix of graph as the feature vector representations for nodes . DeepWalk : If we restrict each component to values of 0 or 1 , the code for each word will be a binary code . In this case , the code learning problem is equivalent to a matrix factorization problem with binary components . Forcing the compact codes to be binary numbers can be beneficial , as the learning problem is usually easier to solve in the binary case , and some existing optimization algorithms in learning to hash can be reused . We should point out that W n p = θ n θ T n is only the first order decomposition with a strong assumption . The general matrix decomposition method can be derived that : In this case , θ i n ∈ R K . While it is straightforward to learn during training , we instead set it to satisfy throughout our experiments , such that the cluster separation ( margin ) in the learned embedding space is three standard deviations . We formulate the mean shift algorithm in a matrix form . Let denote the stacked pixel embedding vectors of an image . A single layer network with non - linearity was adopted in recursive auto - associate memories and recursive autoencoders . socher2012semantic extended this network with an additional matrix representation for each node to augment the expressive power of the model . Tensor networks have also been used as composition function for sentence - level sentiment analysis task . Moreover , the fact that the two components are separately trained can limit its expressiveness . levy2014neural also considered the matrix factorization perspective , but in the context of learning word embeddings . In a general sense , Mixture of Softmaxes proposed in this work can be seen as a particular instantiation of the long - existing idea called Mixture of Experts ( MoE ) jacobs1991adaptive . As a result , ( 8 ) reduces to the following least square optimization : Update ( 9 ) can still be computationally expensive because of the matrix inversion . We can derive a further approximation by performing only one step of gradient descent of ( 8 ) ( or ( 9 ) ) , which gives Only when the buffer is full , or a stack is about to be pruned , the n - grams are scored by the CSLM . This allows us to perform fast matrix - matrix multiplication on GPU using Theano . subsection : Quantitative Analysis The recursive models employ a syntactic parse tree as the sentence structure and the sentence representation is computed recursively in a bottom - up manner along the parse tree . Under this category , we choose recursive autoencoder ( RAE ) , matrix - vector ( MV - RNN ) , tensor based composition ( RNTN ) and multi - layer stacked ( DRNN ) recursive neural network as baselines . Among CNNs , we compare with Kim ’s kim CNN model with fine - tuned word vectors ( CNN - non - static ) and multi - channels ( CNN - multichannel ) , DCNN with dynamic k - max pooling , Tao ’s CNN ( Molding - CNN ) with low - rank tensor based non - linear and non - consecutive convolutions . The control law is designed to be easily implementable , requiring no observers , function approximators , or adaptive laws . By exploiting minimal knowledge of the structure of the nonlinear SJA dynamic model , a matrix decomposition technique is exploited to compensate for the input-multiplicative parametric uncertainty inherent in the SJA dynamics . The control law is shown to yield global asymptotic trajectory tracking in the presence of parametric uncertainty , actuator nonlinearity , and unknown , nonlinear external disturbances . We propose a new technique by extending the decomposition formulation with bounded errors , named Extended LSD ( E-LSD ) . This formulation integrates low-rank background , structured sparse foreground , as well as their residuals in a matrix decomposition problem . Solving this optimization problem is challenging . The wave front method and its improvement are the current resolving method . Based on the analysis of characteristic , improvement and application of wave front method , some improved arithmetic are proposed , such as matrix decomposing , large number processing , node dividing and numbering , calculating factor including , big wave setting . Some improved arithmetic methods are validated by C computing language . In many cases , such approaches more properly utilize the advantageous features of optical systolic array processors . The matrix-decomposition operation ( rather than solution of the simplified matrix-vector equation that results ) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system . The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution . We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming ( QCQP ) . We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral ( piecewise linear ) underestimation . By employing suitable matrix cones , we then show that the convex conic relaxation can be reduced to a semidefinite programming ( SDP ) problem . The other is to embed the global information into the network by appending the low-rank constraint over the network outputs . To verify the ability of matrix decompose , our method is used for image restoration . And to evaluate the structure of low-dimensional space , the latent representation is exploited for cluster . For comparison purposes , the direct and Biglieri quadratic structures are included in this study . Evaluating the comparative results , the conclusion drawn is that the matrix decomposition realization of a quadratic filter implemented with systolic arrays seems to be offering the best compromise among the various conflicting figures of merit . With the recursive inversion processes on the reduced matrix , it is easy to obtain the solutions . Combined with the fast adaptive cross sampling and the matrix decomposition algorithm method , the speed of the calculation of FACA is proved much faster than that of the conventional adaptive cross approximation . The numerical results of several perfect electric conducting objects are validated to demonstrate the performance of this method . In addition , the Feig 's algorithm is extended to other three forms . Utilizing matrix decomposition representation , the links and differences between the two algorithms are revealed . This decomposition representation helps understanding and further improving the algorithms using matrix theory . In QRN , the final reduced queries ( h t ) can be decomposed into computing over candidate reduced queries ( h t ) , without looking at the previous reduced query . Here we primarily show that the query update in Equation 3 can be parallelized by rewriting the equation with matrix operations . The extension to Equation 5 is straightforward . We assume that is a slow enough process . We have a linear update rule for the fast iterate using the vector function and the matrix function . paragraph : Assumptions .
We believe that the techniques used to overcome missing data problem in other domains can also be employed in software engineering . In this paper we propose Matrix Factorization algorithm to tackle with missing data problem in building predictive models in software development domain .
We provide an upper-bound for the bias of the proposed method . Our experiments show that , our method is 40 times as fast as the baseline method on finding the optimal model order in a matrix factorization problem .
Results Watson ’s natural language processing generated a relationship network matching 55 ( 86 % ) genes upstream of BTK and 98 ( 95 % ) genes downstream of Bruton ’s tyrosine kinase in a prominent manually curated physical interaction network . Matrix factorization analysis predicted 11 of 13 genes identified by Merck subject matter experts in the top 20 % of Watson for Drug Discovery ’s 13,595 ranked genes , with 7 in the top 5 % . Conclusion Taken together , these results suggest that Watson for Drug Discovery ’s automatic relationship network identifies the majority of upstream and downstream genes in biological pathway networks and can be used to help with the identification and prioritization of pharmacodynamic biomarker evaluation , accelerating the early phases of disease hypothesis generation .
This has been recently exploited for the fusion of low spatial resolution HSI with high spatial resolution multispectral images in order to obtain super-resolution HSI . Most approaches adopt an unmixing or a matrix factorization perspective . The derived methods have led to state-of-the-art results when the spectral information lies in a low-dimensional subspace/manifold .
This allows us to rank new items for this user . We then enrich the item and user representations in order to perform rating prediction using a classical matrix factorization scheme . We demonstrate the interest of our approach regarding both item ranking and rating prediction on a series of classical benchmarks .
Other than known challenges in collaborative filtering techniques , accuracy and computational cost at a large scale data still at saliency . In this paper we proposed an approach by utilizing matrix value factorization for predicting rating i by user j with the sub matrix as k-most similar items specific to user i for all users who rated them all . In an attempt , previously predicted values are used for subsequent predictions and we have investigated the accuracy of neighborhood methods by applying our method on Netflix Prize ( http://www.netflixprize.com/ ) .
The disadvantage of the traditional CFAbMD algorithm is no consideration of impact of local users ’ neighbor on item rating . Aiming at this problem , a new CFAbMD algorithm is proposed considering both ALS matrix factorization and user nearest neighbor ( CFAbMD-UNN ) , which integrates the similarity information among users into the matrix factorization of model . Furthermore , the CFAbMD-UNN algorithm was implemented in parallel on Spark .
A digital camera with zoom lenses is used to capture images of the investigated object . The image sequence is then transferred to a Matlab ® code to detect the interest points , then to compute the 3-D coordinates of those interest points and the camera poses by matrix factorization operation . Neither calibration object nor a priori information of the camera is needed .
Moreover , users are usually more interested in the top 20 or even top 10 recommended POIs , which makes personalized ranking important in this task . From previous work , directly optimizing for pairwise ranking like Bayesian Personalized Ranking ( BPR ) achieves better performance in the top-k recommendation than directly using matrix matrix factorization that aims to minimize the point-wise rating error . To consider users ’ preferences , geographical influence and personalized ranking , we propose a unified POI recommendation framework , which unifies all of them together .
The updated hidden state vector is then used to predict a probability distribution over the next sequence element , using function . In the case where consists of mutually exclusive discrete outcomes , may apply a matrix multiplication followed by a softmax function : Generative RNNs can evaluate log - likelihoods of sequences exactly , and are differentiable with respect to these log - likelihoods . RNNs can be difficult to train due to the vanishing gradient problem , but advances such as the long short - term memory architecture ( LSTM ) have allowed RNNs to be successful .
Spectral clustering : This is a matrix factorization approach in which we take the top eigenvectors of the normalized Laplacian matrix of graph as the feature vector representations for nodes . DeepWalk :
If we restrict each component to values of 0 or 1 , the code for each word will be a binary code . In this case , the code learning problem is equivalent to a matrix factorization problem with binary components . Forcing the compact codes to be binary numbers can be beneficial , as the learning problem is usually easier to solve in the binary case , and some existing optimization algorithms in learning to hash can be reused .
We should point out that W n p = θ n θ T n is only the first order decomposition with a strong assumption . The general matrix decomposition method can be derived that : In this case , θ i n ∈ R K .
While it is straightforward to learn during training , we instead set it to satisfy throughout our experiments , such that the cluster separation ( margin ) in the learned embedding space is three standard deviations . We formulate the mean shift algorithm in a matrix form . Let denote the stacked pixel embedding vectors of an image .
A single layer network with non - linearity was adopted in recursive auto - associate memories and recursive autoencoders . socher2012semantic extended this network with an additional matrix representation for each node to augment the expressive power of the model . Tensor networks have also been used as composition function for sentence - level sentiment analysis task .
Moreover , the fact that the two components are separately trained can limit its expressiveness . levy2014neural also considered the matrix factorization perspective , but in the context of learning word embeddings . In a general sense , Mixture of Softmaxes proposed in this work can be seen as a particular instantiation of the long - existing idea called Mixture of Experts ( MoE ) jacobs1991adaptive .
As a result , ( 8 ) reduces to the following least square optimization : Update ( 9 ) can still be computationally expensive because of the matrix inversion . We can derive a further approximation by performing only one step of gradient descent of ( 8 ) ( or ( 9 ) ) , which gives
Only when the buffer is full , or a stack is about to be pruned , the n - grams are scored by the CSLM . This allows us to perform fast matrix - matrix multiplication on GPU using Theano . subsection : Quantitative Analysis
The recursive models employ a syntactic parse tree as the sentence structure and the sentence representation is computed recursively in a bottom - up manner along the parse tree . Under this category , we choose recursive autoencoder ( RAE ) , matrix - vector ( MV - RNN ) , tensor based composition ( RNTN ) and multi - layer stacked ( DRNN ) recursive neural network as baselines . Among CNNs , we compare with Kim ’s kim CNN model with fine - tuned word vectors ( CNN - non - static ) and multi - channels ( CNN - multichannel ) , DCNN with dynamic k - max pooling , Tao ’s CNN ( Molding - CNN ) with low - rank tensor based non - linear and non - consecutive convolutions .
The control law is designed to be easily implementable , requiring no observers , function approximators , or adaptive laws . By exploiting minimal knowledge of the structure of the nonlinear SJA dynamic model , a matrix decomposition technique is exploited to compensate for the input-multiplicative parametric uncertainty inherent in the SJA dynamics . The control law is shown to yield global asymptotic trajectory tracking in the presence of parametric uncertainty , actuator nonlinearity , and unknown , nonlinear external disturbances .
We propose a new technique by extending the decomposition formulation with bounded errors , named Extended LSD ( E-LSD ) . This formulation integrates low-rank background , structured sparse foreground , as well as their residuals in a matrix decomposition problem . Solving this optimization problem is challenging .
The wave front method and its improvement are the current resolving method . Based on the analysis of characteristic , improvement and application of wave front method , some improved arithmetic are proposed , such as matrix decomposing , large number processing , node dividing and numbering , calculating factor including , big wave setting . Some improved arithmetic methods are validated by C computing language .
In many cases , such approaches more properly utilize the advantageous features of optical systolic array processors . The matrix-decomposition operation ( rather than solution of the simplified matrix-vector equation that results ) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system . The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution .
We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming ( QCQP ) . We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral ( piecewise linear ) underestimation . By employing suitable matrix cones , we then show that the convex conic relaxation can be reduced to a semidefinite programming ( SDP ) problem .
The other is to embed the global information into the network by appending the low-rank constraint over the network outputs . To verify the ability of matrix decompose , our method is used for image restoration . And to evaluate the structure of low-dimensional space , the latent representation is exploited for cluster .
For comparison purposes , the direct and Biglieri quadratic structures are included in this study . Evaluating the comparative results , the conclusion drawn is that the matrix decomposition realization of a quadratic filter implemented with systolic arrays seems to be offering the best compromise among the various conflicting figures of merit .
With the recursive inversion processes on the reduced matrix , it is easy to obtain the solutions . Combined with the fast adaptive cross sampling and the matrix decomposition algorithm method , the speed of the calculation of FACA is proved much faster than that of the conventional adaptive cross approximation . The numerical results of several perfect electric conducting objects are validated to demonstrate the performance of this method .
In addition , the Feig 's algorithm is extended to other three forms . Utilizing matrix decomposition representation , the links and differences between the two algorithms are revealed . This decomposition representation helps understanding and further improving the algorithms using matrix theory .
In QRN , the final reduced queries ( h t ) can be decomposed into computing over candidate reduced queries ( h t ) , without looking at the previous reduced query . Here we primarily show that the query update in Equation 3 can be parallelized by rewriting the equation with matrix operations . The extension to Equation 5 is straightforward .
We assume that is a slow enough process . We have a linear update rule for the fast iterate using the vector function and the matrix function . paragraph : Assumptions .
Handling missing attributes using matrix factorization Stochastic thermodynamic integration: Efficient Bayesian model selection via stochastic gradient MCMC Identification of pharmacodynamic biomarker hypotheses through literature analysis with IBM Watson Hyperspectral Super-Resolution of Locally Low Rank Images From Complementary Multisource Data Latent Trajectory Modeling: A Light and Efficient Way to Introduce Time in Recommender Systems Movie Rating Prediction with Matrix Factorization Algorithm Collaborative Filtering Recommendation Algorithm Based on Matrix Factorization and User Nearest Neighbors Monocular three-dimensional self-calibrating surface reconstruction using digital photogrammetry A Unified Point-of-Interest Recommendation Framework in Location-Based Social Networks Multiplicative LSTM for sequence modelling node2vec: Scalable Feature Learning for Networks Compressing Word Embeddings via Deep Compositional Code Learning Product-Based Neural Networks for User Response Prediction Recurrent Pixel Embedding for Instance Grouping Neural Tree Indexers for Text Understanding Breaking the Softmax Bottleneck: A High-Rank RNN Language Model Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation A C-LSTM Neural Network for Text Classification Robust nonlinear aircraft tracking control using synthetic jet actuators Error Bounded Foreground and Background Modeling for Moving Object Detection in Satellite Videos Summary and Improvement on Wave Front Method in Numerical Calculation of Finite Element Method with Huge Node Quantity Direct and implicit optical matrix-vector algorithms. Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation Embedded conformal deep low-rank auto-encoder network for matrix recovery Efficient implementations of quadratic digital filters Multiscale Compressed Block Decomposition Method With Characteristic Basis Function Method and Fast Adaptive Cross Approximation The matrix decomposition representation of DCT algorithms Query-Reduction Networks for Question Answering GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium
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RAISE@ICSE ICASSP RecSys The Influence of Technology on Social Network Analysis and Mining AsiaSim/SCS AutumnSim (1) ICLR KDD ICLR 2016 IEEE 16th International Conference on Data Mining (ICDM) 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition EACL ICLR ArXiv EMNLP ArXiv CDC ICLR NIPS
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