In the experiments , we evaluate our method on the tasks of link prediction and triplet classification . The experimental results indicate that the Wasserstein distance is capable of solving the problem of vanishing gradient on discrete data and accelerating the convergence , additional weak supervision information also can significantly improve the performance of the model . The model determines the location and size of Electric Vehicle Charging Stations ( EVCSs ) , Renewable Energy Sources ( RESs ) , Battery Energy Storage System ( BESSs ) , and distribution network expansion schemes , with the objectives of minimizing the total investment and reliability cost of the distribution network and maximizing the EVCSs ’ charging service capability . A scenario-based stochastic modelling approach based on Wasserstein distance metric and K-medoids scenario analysis is developed to model the stochastic nature of renewable generation . A multi-objective optimization algorithm , Multi-Objective Natural Aggregation Algorithm ( MONAA ) , is applied to solve the proposed model . A Monte Carlo Simulation.- Finding Meaningful and Stable Clusters Using Local Cluster Analysis.- Comparing Optimal Individual and Collective Assessment Procedures.- Network and Graph Analysis.- Some Open Problem Sets for Generalized Blockmodeling.- Spectral Clustering and Multidimensional Scaling : A Unified View.- Analyzing the Structure of U.S. Patents Network.- Identifying and Classifying Social Groups : A Machine Learning Approach.- Analysis of Symbolic Data.- Multidimensional Scaling of Histogram Dissimilarities.- Dependence and Interdependence Analysis for Interval-Valued Variables.- A New Wasserstein Based Distance for the Hierarchical Clustering of Histogram Symbolic Data.- Symbolic Clustering of Large Datasets.- A Dynamic Clustering Method for Mixed Feature-Type Symbolic Data.- General Data Analysis Methods.- Iterated Boosting for Outlier Detection.- Sub-species of Homopus Areolatus ? Biplots and Small Class Inference with Analysis of Distance.- Revised Boxplot Based Discretization as the Kernel of Automatic Interpretation of Classes Using Numerical Variables.- Data and Web Mining.- Comparison of Two Methods for Detecting and Correcting Systematic Error in High-throughput Screening Data.- kNN Versus SVM in the Collaborative Filtering Framework.- Mining Association Rules in Folksonomies.- Empirical Analysis of Attribute-Aware Recommendation Algorithms with Variable Synthetic Data.- Patterns of Associations in Finite Sets of Items.- Analysis of Music Data.- Generalized N-gram Measures for Melodic Similarity.- Evaluating Different Approaches to Measuring the Similarity of Melodies.- Using MCMC as a Stochastic Optimization Procedure for Musical Time Series.- Local Models in Register Classification by Timbre.- Gene and Microarray Analysis.- Improving the Performance of Principal Components for Classification of Gene Expression Data Through Feature Selection.- A New Efficient Method for Assessing Missing Nucleotides in DNA Sequences in the Framework of a Generic Evolutionary Model.- New Efficient Algorithm for Modeling Partial and Complete Gene Transfer Scenarios . First , we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix . We then leverage this accurate moment recovery , via a Wasserstein distance argument , to show that the vector of eigenvalues can be accurately recovered . We provide finite -- sample bounds on the expected error of the recovered eigenvalues , which imply that our estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity , even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero . The tabu search artificial bee colony algorithm is used to solve the optimization problems . The simulation results show that the optimal scenes based on the Wasserstein distance indicators and K-means cluster technique reflect the random feature of distributed generation active power output more accurately . Finally , the simulation analysis of IEEE33-bus distribution test system is carried out to verify the effectiveness and feasibility of the proposed method . To analyze the actual bidding behavior in power markets , this paper proposes a data-driven analysis framework for bidding behavior in which a data standardization processing method is proposed that addresses the particularities of the bidding data and provides a fundamental dataset for further market analyses . Then , an adaptive clustering method for bidding behavior is developed that applies the $ {K}$ -medoids method and the Wasserstein distance measurement to extract the generators ’ bidding patterns from a massive dataset . An empirical analysis of the bidding behavior is conducted on actual data from the Australian energy market . COLOR CHILD is shown to outperform all of the known state-of-the-art image local descriptors of parametric and non-parametric types on a variety of benchmark texture databases such as KTH-TIPS2-a , KTH-TIPS2-b , and CUReT under varying degrees of noise while performing texture classification task . Further , the response profile of the COLOR CHILD in terms of Wasserstein distance measures ( obtained by sliding a query patch across the image to be segmented ) is found to be better suited as initial image for active contour-based image and texture segmentation algorithms . The efficacy of the COLOR CHILD for segmentation task is amply demonstrated on synthetic color images under varying degrees of noise and on real-world texture images . [ reference ] . This approach has recently been shown to yield locally convergent optimization in some cases with non - continuous distributions , where the original GAN does not [ reference ] . In this paper , we introduce a novel regularization for the MMD GAN critic of [ reference ][ reference ][ reference ] , which directly targets generator performance , rather than adopting regularization methods intended to approximate Wasserstein distances [ reference ][ reference ] . [ reference ] . This could be because for some kernel classes , D Ψ MMD is stronger than Wasserstein or MMD . Example 1 Another area to explore is the geometry of these losses , as studied by Bottou et al . [ reference ] , who showed potential advantages of the Wasserstein geometry over the MMD . Their results , though , do not address any distances based on optimized kernels ; the new distances introduced here might have interesting geometry of their own . P with finite moment follows directly from Assumptions ( A ) and ( B ) , since section : A.2 Continuity of the Optimized Scaled MMD in the Wasserstein topology To prove Theorem 1 , we we will first need some new notation . This study focuses on image generation tasks using convolutional neural networks ( CNN ) for both and . Several loss functions have been proposed to quantify the difference of the scores between real and generated samples : and , including the minimax loss and non - saturating loss ( ) , hinge loss ( ) , Wasserstein loss ( ) and maximum mean discrepancy ( MMD ) ( ) ( see Appendix [ reference ] for more details ) . If is injective , is characteristic and reaches its minimum if and only if ( ) . Thus , the objective functions for and could be ( ) : MMD - GAN has been shown to be more effective than the model that directly uses MMD as the loss function for the generator ( ) . showed that MMD and Wasserstein metric are weaker objective functions for GAN than the JensenâShannon ( JS ) divergence ( related to minimax loss ) and total variation ( TV ) distance ( related to hinge loss ) . The reason is that convergence of to in JS - divergence and TV distance also implies convergence in MMD and Wasserstein metric . The Gaussian is the maximum entropy distribution for given mean and covariance , therefore we assume the coding units to follow a multidimensional Gaussian . The difference of two Gaussians ( synthetic and real - world images ) is measured by the Fréchet distance also known as Wasserstein - 2 distance . We call the Fréchet distance between the Gaussian with mean obtained from and the Gaussian with mean obtained from the ‘ ‘ We employ the local RT by aligning the centre of the RT with the centroids of the region of interest and use a sufficient number of projections . Finally the earth mover 's distance method is utilized to combine local matching results . Using the proposed approach , image retrieval accuracy is maintained , while reducing computational cost . The compression efficiency was measured as a function of image brightness ( and SNR ) . Furthermore , the alterations introduced by compression were estimated using brightness histograms ( earth ’s mover distance algorithm ) and textures ( Haralick parameters ) . Second , we introduce two algorithms to compare transition graphs . The first is the classical earth mover distance metric which depends only on local minimum energy configurations along with their statistical weights , while the second incorporates topological constraints inherent to conformational transitions . Illustrations are provided on a simplified protein model ( BLN69 ) , whose frustrated potential energy landscape has been thoroughly studied . We give a partial answer to this problem , by showing that for norms that admit efficient linear sketches , the optimal number of measurements $ m$ is closely related to the doubling dimension of the metric induced by the norm $ \|\cdot\|$ on the set of all $ k$-sparse vectors . By applying our result to specific norms , we cast known measurement bounds in our general framework ( for the $ \ell_p$ norms , $ p \in [1,2]$ ) as well as provide new , measurement-efficient schemes ( for the Earth-Mover Distance norm ) . The latter result directly implies more succinct linear sketches for the well-studied planar $ k$-median clustering problem . Color segmentation clusters regions of similar color in each frame . A novel , likelihood ratio-based method for the statistical comparison of color layers in the regions of activity and the background is presented and compared with an Earth mover 's distance-based approach . Our method also gives the likelihood with which each pixel is assigned to a moving object in each frame . We measure distance between trees using the best possible super-imposition that minimizes the number of mismatched leaf labels . Our distance measure is equivalent to an Earth Mover 's Distance measure since the leaf-labeled trees of height one can be viewed as sets and can be recursively extended to trees of larger height by viewing them as set of sets . We compute sketches of arbitrary weighted trees and analyze them in the context of locality-sensitive hashing ( LSH ) where the probability of two sketches matching is high when two trees are similar and low when the two trees are far under the given distance measure . and S gt i are the prediction and groundtruth of each sample , respectively . We propose two candidates : Chamfer distance ( CD ) and Earth Mover 's distance ( EMD ) [ reference ] . The exact computation of the goal direction would require knowledge about the distances to visible landmarks , information , which is not directly available to passive vision systems . However , if prior assumptions about typical distance distributions are used , a snapshot taken at the goal suffices to compute the goal direction from the current view . We show that most existing approaches to scene-based homing implicitly assume an isotropic landmark distribution . Based on such single-valued distance functions OPTICS , like other standard data mining algorithms , can work without any changes . In this paper , we propose to express the similarity between two fuzzy objects by distance probability functions which assign a probability value to each possible distance value . Contrary to the traditional approach , we do not extract aggregated values from the fuzzy distance functions but enhance OPTICS so that it can exploit the full information provided by these functions . In our paper , we point out cases where a direct application of sparse coding will lead to a failure of knowledge transfer . We then present the details of our extension to sparse coding , by incorporating distribution distance estimates for the embedded data , and show that the proposed algorithm can overcome the shortcomings of the sparse coding algorithm on synthetic data and achieve improved predictive performance on a real world chemical toxicity transfer learning task . We first discuss domain adaptation ( DA ) techniques that rely on the Maximum Mean Discrepancy to measure such a distance . We then study the use of alternative distribution distance measures within one specific Domain Adaptation framework . In this context , we focus on f-divergences , and in particular on the KL divergence and the Hellinger distance . In this paper , we discussed the problem of combining prior distributions based on Kullback information . Using the Kullback information as the distance measure of distributions , the criterions minimized the sum of the distance between the prior distributions and the aggregated distribution . An example is given to illuminate this method . First we argue for a set of desirable properties , review commonly used measures , and conclude that no single measure unites all desirable properties . However the area under the ROC curve ( a classification measure ) and the KL-divergence ( a distance measure of probability distributions ) combine many desirable properties and allow a meaningful comparison of critical model performance . We give an analytical proof of the linearity of the ROC measure with respect to averaging over subjects and demonstrate an appropriate correction of entropy-based measures like KL-divergence for small sample sizes in the context of eye-tracking data . O ne of the important issues in everyday life is optimization problem involving reduction of cost and distance of distribution and transportation of goods . Researchers are always in line with this objective , by providing search tools trying various approaches to minimize such costs and distances . Using expectation maximization algorithm , we construct a mixture model of von Mises distribution describing the set of directions , and a mixture model of normal distribution related to the speed set . Each frame is compared with a collection of reference frames using distance of probability densities . This distance is then used to indicate changes in the crowd motion . Learning strategies considered are writer independent- those that learn from a set of signature sample ( including forgeries ) prior to enrollment of a writer , and writer dependent- those that learn only from a newly enrolled individual . Classification methods considered include two distance based methods ( one based on a threshold , which is the standard method of signature verification and biometrics , and the other based on a distance probability distribution ) , a Nave Bayes ( NB ) classifier based on pairs of feature bit values and a support vector machine ( SVM ) . Two scenarios are considered for the writer dependent scenario : ( i ) without forgeries ( one-class problem ) and ( ii ) with forgery samples being available ( two class problem ) . Based on the theoretical result , we present some meaningful results on the finite three-dimensional network performance , including the node degree and the max-flow capacity . Furthermore , we investigate some approximation properties of the distance probability distribution function derived in the paper . A second-order stochastic model describing a large scale crowd is formulated , and an efficient evacuation strategy for agents in complex surroundings is proposed and solved numerically . The method consists in reshaping the crowd contour by making use of the crowd density feedback that is commonly available from geolocation technologies , and Kantorovich distance is used to transport the current shape into the desired one . The availability of the crowd density enables to solve the otherwise challenging forward-backward problem . In addition , we show that any algorithm that approximates the distance within a certain absolute error , depending on the size of the SMMs , is NP-hard . Nevertheless , we introduce an alternative distance , based on the Kantorovich metric , that is an over-approximation of the former and we show that , under mild assumptions on the residence time distributions , it can be computed in polynomial time . A key ingredient of our algorithm is Tarski 's decision procedure for the first order theory over real closed fields . By exploiting the Kantorovich-Rubinstein duality theorem we can restrict to the existential fragment for which more efficient decision procedures exist . The Hellinger-Kantorovich Distance and Geodesic Curves . We discuss a new notion of distance on the space of finite and nonnegative measures on $ \Omega \subset { \mathbb R}^d$ , which we call the Hellinger -- Kantorovich distance . It can be seen as an inf-convolution of the well-known Kantorovich -- Wasserstein distance and the Hellinger-Kakutani distance . The goals of this chapter are to : Introduce the Kantorovich and Kantorovich – Rubinstein problems in one-dimensional and multidimensional settings ; Provide examples illustrating applications of the abstract problems ; Provide examples illustrating applications of the abstract problems ; Discuss the multivariate Kantorovich and Kantorovich – Rubinstein theorems , which provide dual representations of certain types of minimal distances and norms ; Discuss a particular application leading to an explicit representation for a class of minimal norms . We combine this regularizer with a matching Wasserstein data fidelity term . Using the Kantorovich-Rubinstein duality , the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm . We demonstrate the effectiveness of the proposed framework on real and synthetic Q-ball data . Both the weighted and unweighted UniFrac distances have been very successfully employed to assess if two communities differ , but do not give any information about how two communities differ . We take advantage of recent observations that the UniFrac metric is equivalent to the so-called earth mover ’s distance ( also known as the Kantorovich – Rubinstein metric ) to develop an algorithm that not only computes the UniFrac distance in linear time and space , but also simultaneously finds which operational taxonomic units are responsible for the observed differences between samples . This allows the algorithm , called EMDUniFrac , to determine why given samples are different , not just if they are different , and with no added computational burden . We further prove that for norms that are closed under sum-product , efficient sketching is equivalent to embedding into l1 with constant distortion . Examples of such norms include the Earth Mover 's Distance ( specifically its norm variant , called Kantorovich-Rubinstein norm ) , and the trace norm ( a.k.a . Schatten 1-norm or the nuclear norm ) . All constructions are described as the special case of a unique one , which we call the Hutchinson hemi-metric on various spaces of continuous previsions . We show a directed form of the Kantorovich-Rubinstein theorem , stating that the Hutchinson hemi-metric on spaces of continuous probability valuations coincides with a notion of trans-shipment hemi-metric . We also identify the class of so-called sym-compact spaces as the right class of topological spaces , where the theory works out as nicely as possible . Under this regularity condition , we have proved in Theorem [ reference ] that the density of generated samples matches the underlying data density . On the contrary , the WGAN introduces the Lipschitz constraint from the Kantorovich - Rubinstein duality of the EM distance but it is not proved in if the density of samples generated by WGAN is consistent with that of real data . Here we assert that the WGAN also models an underlying Lipschitz density .
In the experiments , we evaluate our method on the tasks of link prediction and triplet classification . The experimental results indicate that the Wasserstein distance is capable of solving the problem of vanishing gradient on discrete data and accelerating the convergence , additional weak supervision information also can significantly improve the performance of the model .
The model determines the location and size of Electric Vehicle Charging Stations ( EVCSs ) , Renewable Energy Sources ( RESs ) , Battery Energy Storage System ( BESSs ) , and distribution network expansion schemes , with the objectives of minimizing the total investment and reliability cost of the distribution network and maximizing the EVCSs ’ charging service capability . A scenario-based stochastic modelling approach based on Wasserstein distance metric and K-medoids scenario analysis is developed to model the stochastic nature of renewable generation . A multi-objective optimization algorithm , Multi-Objective Natural Aggregation Algorithm ( MONAA ) , is applied to solve the proposed model .
A Monte Carlo Simulation.- Finding Meaningful and Stable Clusters Using Local Cluster Analysis.- Comparing Optimal Individual and Collective Assessment Procedures.- Network and Graph Analysis.- Some Open Problem Sets for Generalized Blockmodeling.- Spectral Clustering and Multidimensional Scaling : A Unified View.- Analyzing the Structure of U.S. Patents Network.- Identifying and Classifying Social Groups : A Machine Learning Approach.- Analysis of Symbolic Data.- Multidimensional Scaling of Histogram Dissimilarities.- Dependence and Interdependence Analysis for Interval-Valued Variables.- A New Wasserstein Based Distance for the Hierarchical Clustering of Histogram Symbolic Data.- Symbolic Clustering of Large Datasets.- A Dynamic Clustering Method for Mixed Feature-Type Symbolic Data.- General Data Analysis Methods.- Iterated Boosting for Outlier Detection.- Sub-species of Homopus Areolatus ? Biplots and Small Class Inference with Analysis of Distance.- Revised Boxplot Based Discretization as the Kernel of Automatic Interpretation of Classes Using Numerical Variables.- Data and Web Mining.- Comparison of Two Methods for Detecting and Correcting Systematic Error in High-throughput Screening Data.- kNN Versus SVM in the Collaborative Filtering Framework.- Mining Association Rules in Folksonomies.- Empirical Analysis of Attribute-Aware Recommendation Algorithms with Variable Synthetic Data.- Patterns of Associations in Finite Sets of Items.- Analysis of Music Data.- Generalized N-gram Measures for Melodic Similarity.- Evaluating Different Approaches to Measuring the Similarity of Melodies.- Using MCMC as a Stochastic Optimization Procedure for Musical Time Series.- Local Models in Register Classification by Timbre.- Gene and Microarray Analysis.- Improving the Performance of Principal Components for Classification of Gene Expression Data Through Feature Selection.- A New Efficient Method for Assessing Missing Nucleotides in DNA Sequences in the Framework of a Generic Evolutionary Model.- New Efficient Algorithm for Modeling Partial and Complete Gene Transfer Scenarios .
First , we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix . We then leverage this accurate moment recovery , via a Wasserstein distance argument , to show that the vector of eigenvalues can be accurately recovered . We provide finite -- sample bounds on the expected error of the recovered eigenvalues , which imply that our estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity , even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero .
The tabu search artificial bee colony algorithm is used to solve the optimization problems . The simulation results show that the optimal scenes based on the Wasserstein distance indicators and K-means cluster technique reflect the random feature of distributed generation active power output more accurately . Finally , the simulation analysis of IEEE33-bus distribution test system is carried out to verify the effectiveness and feasibility of the proposed method .
To analyze the actual bidding behavior in power markets , this paper proposes a data-driven analysis framework for bidding behavior in which a data standardization processing method is proposed that addresses the particularities of the bidding data and provides a fundamental dataset for further market analyses . Then , an adaptive clustering method for bidding behavior is developed that applies the $ {K}$ -medoids method and the Wasserstein distance measurement to extract the generators ’ bidding patterns from a massive dataset . An empirical analysis of the bidding behavior is conducted on actual data from the Australian energy market .
COLOR CHILD is shown to outperform all of the known state-of-the-art image local descriptors of parametric and non-parametric types on a variety of benchmark texture databases such as KTH-TIPS2-a , KTH-TIPS2-b , and CUReT under varying degrees of noise while performing texture classification task . Further , the response profile of the COLOR CHILD in terms of Wasserstein distance measures ( obtained by sliding a query patch across the image to be segmented ) is found to be better suited as initial image for active contour-based image and texture segmentation algorithms . The efficacy of the COLOR CHILD for segmentation task is amply demonstrated on synthetic color images under varying degrees of noise and on real-world texture images .
[ reference ] . This approach has recently been shown to yield locally convergent optimization in some cases with non - continuous distributions , where the original GAN does not [ reference ] . In this paper , we introduce a novel regularization for the MMD GAN critic of [ reference ][ reference ][ reference ] , which directly targets generator performance , rather than adopting regularization methods intended to approximate Wasserstein distances [ reference ][ reference ] .
[ reference ] . This could be because for some kernel classes , D Ψ MMD is stronger than Wasserstein or MMD . Example 1
Another area to explore is the geometry of these losses , as studied by Bottou et al . [ reference ] , who showed potential advantages of the Wasserstein geometry over the MMD . Their results , though , do not address any distances based on optimized kernels ; the new distances introduced here might have interesting geometry of their own .
P with finite moment follows directly from Assumptions ( A ) and ( B ) , since section : A.2 Continuity of the Optimized Scaled MMD in the Wasserstein topology To prove Theorem 1 , we we will first need some new notation .
This study focuses on image generation tasks using convolutional neural networks ( CNN ) for both and . Several loss functions have been proposed to quantify the difference of the scores between real and generated samples : and , including the minimax loss and non - saturating loss ( ) , hinge loss ( ) , Wasserstein loss ( ) and maximum mean discrepancy ( MMD ) ( ) ( see Appendix [ reference ] for more details ) .
If is injective , is characteristic and reaches its minimum if and only if ( ) . Thus , the objective functions for and could be ( ) : MMD - GAN has been shown to be more effective than the model that directly uses MMD as the loss function for the generator ( ) . showed that MMD and Wasserstein metric are weaker objective functions for GAN than the JensenâShannon ( JS ) divergence ( related to minimax loss ) and total variation ( TV ) distance ( related to hinge loss ) . The reason is that convergence of to in JS - divergence and TV distance also implies convergence in MMD and Wasserstein metric .
The Gaussian is the maximum entropy distribution for given mean and covariance , therefore we assume the coding units to follow a multidimensional Gaussian . The difference of two Gaussians ( synthetic and real - world images ) is measured by the Fréchet distance also known as Wasserstein - 2 distance . We call the Fréchet distance between the Gaussian with mean obtained from and the Gaussian with mean obtained from the ‘ ‘
We employ the local RT by aligning the centre of the RT with the centroids of the region of interest and use a sufficient number of projections . Finally the earth mover 's distance method is utilized to combine local matching results . Using the proposed approach , image retrieval accuracy is maintained , while reducing computational cost .
The compression efficiency was measured as a function of image brightness ( and SNR ) . Furthermore , the alterations introduced by compression were estimated using brightness histograms ( earth ’s mover distance algorithm ) and textures ( Haralick parameters ) .
Second , we introduce two algorithms to compare transition graphs . The first is the classical earth mover distance metric which depends only on local minimum energy configurations along with their statistical weights , while the second incorporates topological constraints inherent to conformational transitions . Illustrations are provided on a simplified protein model ( BLN69 ) , whose frustrated potential energy landscape has been thoroughly studied .
We give a partial answer to this problem , by showing that for norms that admit efficient linear sketches , the optimal number of measurements $ m$ is closely related to the doubling dimension of the metric induced by the norm $ \|\cdot\|$ on the set of all $ k$-sparse vectors . By applying our result to specific norms , we cast known measurement bounds in our general framework ( for the $ \ell_p$ norms , $ p \in [1,2]$ ) as well as provide new , measurement-efficient schemes ( for the Earth-Mover Distance norm ) . The latter result directly implies more succinct linear sketches for the well-studied planar $ k$-median clustering problem .
Color segmentation clusters regions of similar color in each frame . A novel , likelihood ratio-based method for the statistical comparison of color layers in the regions of activity and the background is presented and compared with an Earth mover 's distance-based approach . Our method also gives the likelihood with which each pixel is assigned to a moving object in each frame .
We measure distance between trees using the best possible super-imposition that minimizes the number of mismatched leaf labels . Our distance measure is equivalent to an Earth Mover 's Distance measure since the leaf-labeled trees of height one can be viewed as sets and can be recursively extended to trees of larger height by viewing them as set of sets . We compute sketches of arbitrary weighted trees and analyze them in the context of locality-sensitive hashing ( LSH ) where the probability of two sketches matching is high when two trees are similar and low when the two trees are far under the given distance measure .
and S gt i are the prediction and groundtruth of each sample , respectively . We propose two candidates : Chamfer distance ( CD ) and Earth Mover 's distance ( EMD ) [ reference ] .
The exact computation of the goal direction would require knowledge about the distances to visible landmarks , information , which is not directly available to passive vision systems . However , if prior assumptions about typical distance distributions are used , a snapshot taken at the goal suffices to compute the goal direction from the current view . We show that most existing approaches to scene-based homing implicitly assume an isotropic landmark distribution .
Based on such single-valued distance functions OPTICS , like other standard data mining algorithms , can work without any changes . In this paper , we propose to express the similarity between two fuzzy objects by distance probability functions which assign a probability value to each possible distance value . Contrary to the traditional approach , we do not extract aggregated values from the fuzzy distance functions but enhance OPTICS so that it can exploit the full information provided by these functions .
In our paper , we point out cases where a direct application of sparse coding will lead to a failure of knowledge transfer . We then present the details of our extension to sparse coding , by incorporating distribution distance estimates for the embedded data , and show that the proposed algorithm can overcome the shortcomings of the sparse coding algorithm on synthetic data and achieve improved predictive performance on a real world chemical toxicity transfer learning task .
We first discuss domain adaptation ( DA ) techniques that rely on the Maximum Mean Discrepancy to measure such a distance . We then study the use of alternative distribution distance measures within one specific Domain Adaptation framework . In this context , we focus on f-divergences , and in particular on the KL divergence and the Hellinger distance .
In this paper , we discussed the problem of combining prior distributions based on Kullback information . Using the Kullback information as the distance measure of distributions , the criterions minimized the sum of the distance between the prior distributions and the aggregated distribution . An example is given to illuminate this method .
First we argue for a set of desirable properties , review commonly used measures , and conclude that no single measure unites all desirable properties . However the area under the ROC curve ( a classification measure ) and the KL-divergence ( a distance measure of probability distributions ) combine many desirable properties and allow a meaningful comparison of critical model performance . We give an analytical proof of the linearity of the ROC measure with respect to averaging over subjects and demonstrate an appropriate correction of entropy-based measures like KL-divergence for small sample sizes in the context of eye-tracking data .
O ne of the important issues in everyday life is optimization problem involving reduction of cost and distance of distribution and transportation of goods . Researchers are always in line with this objective , by providing search tools trying various approaches to minimize such costs and distances .
Using expectation maximization algorithm , we construct a mixture model of von Mises distribution describing the set of directions , and a mixture model of normal distribution related to the speed set . Each frame is compared with a collection of reference frames using distance of probability densities . This distance is then used to indicate changes in the crowd motion .
Learning strategies considered are writer independent- those that learn from a set of signature sample ( including forgeries ) prior to enrollment of a writer , and writer dependent- those that learn only from a newly enrolled individual . Classification methods considered include two distance based methods ( one based on a threshold , which is the standard method of signature verification and biometrics , and the other based on a distance probability distribution ) , a Nave Bayes ( NB ) classifier based on pairs of feature bit values and a support vector machine ( SVM ) . Two scenarios are considered for the writer dependent scenario : ( i ) without forgeries ( one-class problem ) and ( ii ) with forgery samples being available ( two class problem ) .
Based on the theoretical result , we present some meaningful results on the finite three-dimensional network performance , including the node degree and the max-flow capacity . Furthermore , we investigate some approximation properties of the distance probability distribution function derived in the paper .
A second-order stochastic model describing a large scale crowd is formulated , and an efficient evacuation strategy for agents in complex surroundings is proposed and solved numerically . The method consists in reshaping the crowd contour by making use of the crowd density feedback that is commonly available from geolocation technologies , and Kantorovich distance is used to transport the current shape into the desired one . The availability of the crowd density enables to solve the otherwise challenging forward-backward problem .
In addition , we show that any algorithm that approximates the distance within a certain absolute error , depending on the size of the SMMs , is NP-hard . Nevertheless , we introduce an alternative distance , based on the Kantorovich metric , that is an over-approximation of the former and we show that , under mild assumptions on the residence time distributions , it can be computed in polynomial time .
A key ingredient of our algorithm is Tarski 's decision procedure for the first order theory over real closed fields . By exploiting the Kantorovich-Rubinstein duality theorem we can restrict to the existential fragment for which more efficient decision procedures exist .
The Hellinger-Kantorovich Distance and Geodesic Curves . We discuss a new notion of distance on the space of finite and nonnegative measures on $ \Omega \subset { \mathbb R}^d$ , which we call the Hellinger -- Kantorovich distance . It can be seen as an inf-convolution of the well-known Kantorovich -- Wasserstein distance and the Hellinger-Kakutani distance .
The goals of this chapter are to : Introduce the Kantorovich and Kantorovich – Rubinstein problems in one-dimensional and multidimensional settings ; Provide examples illustrating applications of the abstract problems ; Provide examples illustrating applications of the abstract problems ; Discuss the multivariate Kantorovich and Kantorovich – Rubinstein theorems , which provide dual representations of certain types of minimal distances and norms ; Discuss a particular application leading to an explicit representation for a class of minimal norms .
We combine this regularizer with a matching Wasserstein data fidelity term . Using the Kantorovich-Rubinstein duality , the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm . We demonstrate the effectiveness of the proposed framework on real and synthetic Q-ball data .
Both the weighted and unweighted UniFrac distances have been very successfully employed to assess if two communities differ , but do not give any information about how two communities differ . We take advantage of recent observations that the UniFrac metric is equivalent to the so-called earth mover ’s distance ( also known as the Kantorovich – Rubinstein metric ) to develop an algorithm that not only computes the UniFrac distance in linear time and space , but also simultaneously finds which operational taxonomic units are responsible for the observed differences between samples . This allows the algorithm , called EMDUniFrac , to determine why given samples are different , not just if they are different , and with no added computational burden .
We further prove that for norms that are closed under sum-product , efficient sketching is equivalent to embedding into l1 with constant distortion . Examples of such norms include the Earth Mover 's Distance ( specifically its norm variant , called Kantorovich-Rubinstein norm ) , and the trace norm ( a.k.a . Schatten 1-norm or the nuclear norm ) .
All constructions are described as the special case of a unique one , which we call the Hutchinson hemi-metric on various spaces of continuous previsions . We show a directed form of the Kantorovich-Rubinstein theorem , stating that the Hutchinson hemi-metric on spaces of continuous probability valuations coincides with a notion of trans-shipment hemi-metric . We also identify the class of so-called sym-compact spaces as the right class of topological spaces , where the theory works out as nicely as possible .
Under this regularity condition , we have proved in Theorem [ reference ] that the density of generated samples matches the underlying data density . On the contrary , the WGAN introduces the Lipschitz constraint from the Kantorovich - Rubinstein duality of the EM distance but it is not proved in if the density of samples generated by WGAN is consistent with that of real data . Here we assert that the WGAN also models an underlying Lipschitz density .
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