Statistical Aspects of Wasserstein Distances

Panaretos, Victor M.; Zemel, Yoav

doi:10.1146/annurev-statistics-030718-104938

Cited by 375 publications

(193 citation statements)

References 118 publications

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“…On the other hand, the Wasserstein-Kantorovich-Rubinstein distance, also known as the "Earth Mover's distance" in the engineering literature, has been used extensively in probability and pattern recognition (e.g., [62][63][64] and references therein). The Wasserstein distance can be interpreted as the cost of transporting a measure from one metric space to a second measure defined on a second metric space; the cost increases with the distance between the metric spaces and the proportion of the measure that needs to be transported.…”

Section: 25mentioning

confidence: 99%

Metrics for graph comparison: A practitioner’s guide

2020

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Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and comparison of structures such as modular communities, rich clubs, hubs, and trees yield insight into the generative mechanisms and functional properties of the graph. Often, two graphs are compared via a pairwise distance measure, with a small distance indicating structural similarity and vice versa. Common choices include spectral distances and distances based on node affinities. However, there has of yet been no comparative study of the efficacy of these distance measures in discerning between common graph topologies at different structural scales. In this work, we compare commonly used graph metrics and distance measures, and demonstrate their ability to discern between common topological features found in both random graph models and real world networks. We put forward a multi-scale picture of graph structure wherein we study the effect of global and local structures on changes in distance measures. We make recommendations on the applicability of different distance measures to the analysis of empirical graph data based on this multi-scale view. Finally, we introduce the Python library NetComp that implements the graph distances used in this work. OPEN ACCESSCitation: Wills P, Meyer FG (2020) Metrics for graph comparison: A practitioner's guide. PLoS ONE 15(2): e0228728. https://doi.org/10.

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Section: 25mentioning

confidence: 99%

Metrics for graph comparison: A practitioner’s guide

2020

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“…Cuturi (), Srivastava et al . (), Sommerfeld and Munk () and Panaretos and Zemel ()). We shall show that the resulting ABC posterior, which we term the Wasserstein ABC (WABC) posterior, can approximate the posterior distribution arbitrarily well in the limit of the threshold ɛ going to 0, while bypassing the choice of summaries.…”

Section: Introductionmentioning

confidence: 99%

Approximate Bayesian Computation with the Wasserstein Distance

Bernton

Jacob

Gerber

et al. 2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

104

145

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Summary A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation has become a popular approach to overcome this issue, in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values. We propose to avoid the use of summaries and the ensuing loss of information by instead using the Wasserstein distance between the empirical distributions of the observed and synthetic data. This generalizes the well‐known approach of using order statistics within approximate Bayesian computation to arbitrary dimensions. We describe how recently developed approximations of the Wasserstein distance allow the method to scale to realistic data sizes, and we propose a new distance based on the Hilbert space filling curve. We provide a theoretical study of the method proposed, describing consistency as the threshold goes to 0 while the observations are kept fixed, and concentration properties as the number of observations grows. Various extensions to time series data are discussed. The approach is illustrated on various examples, including univariate and multivariate g‐and‐k distributions, a toggle switch model from systems biology, a queuing model and a Lévy‐driven stochastic volatility model.

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“…It will often be based on practical considerations such as the feasibility of computations and the interpretability of results. For example, in the case of distributional data, the Wasserstein metric has emerged as a favourite when working with one-dimensional distributions (Bolstad et al, 2003;Zhang and Müller, 2011;Panaretos and Zemel, 2019;Bigot, 2019); however it is not clear that it is the universally best choice for samples of trajectories of random distributions.…”

Section: Flexibility Through Metric Selectionmentioning

confidence: 99%

Functional Models for Time-Varying Random Objects

Dubey

Müller

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued. In recent years, samples of time‐varying object data such as time‐varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. We propose metric covariance, a novel association measure for paired object data lying in a metric space (Ω,d) that we use to define a metric autocovariance function for a sample of random Ω‐valued curves, where Ω generally will not have a vector space or manifold structure. The proposed metric autocovariance function is non‐negative definite when the squared semimetric d2 is of negative type. Then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for Ω‐valued functional data, including time‐varying probability distributions, covariance matrices and time dynamic networks. Analogues of functional principal components for time‐varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real‐valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space Ω. We establish asymptotic consistency of the sample‐based estimators for the corresponding population targets under mild metric entropy conditions on Ω and continuity of the Ω‐valued random curves. These concepts are illustrated with samples of time‐varying probability distributions for human mortality, time‐varying covariance matrices derived from trading patterns and time‐varying networks that arise from New York taxi trips.

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Statistical Aspects of Wasserstein Distances

Cited by 375 publications

References 118 publications

Metrics for graph comparison: A practitioner’s guide

Metrics for graph comparison: A practitioner’s guide

Approximate Bayesian Computation with the Wasserstein Distance

Functional Models for Time-Varying Random Objects

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