On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

By Aaditya Ramdas, Nicolás García Trillos, Marco Cuturi. Entropy, 2015

613 citations recorded by the source corpus.

Nonparametric two sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being intelligently designed and analyzed, both for the unidimensional and the multivariate setting. Our contribution is to tie together many of these tests, drawing connections between seemingly very different statistics. In this work, our central object is the Wasserstein distance, as we form a chain of connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ plots and ROC/ODC curves, to multivariate tests involving energy statistics and kernel based maximum mean discrepancy. Some connections proceed through the construction of a \textit{smoothed} Wasserstein distance, and others through the pursuit of a "distribution-free" Wasserstein test. Some observations in this chain are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two sample testing's classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0fa622b2ee89456ff75235fa0bcc642c09076ebd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Statistics and Probability

Citations in the corpus

Optimal Mass Transport: Signal processing and machine-learning applications (2017, methodology): “57 IEEE SIgnal ProcESSIng MagazInE | July 2017 | inference [12], [13] and hypothesis testing [15], [42] among others.”
Hybrid Random Features (2021, methodology): “We compute the 1 -dimensional Wasserstein distance and the Kolmogrov-Smirnov (KS) metric (Ramdas et al., 2017) between the approximated softmax distribution and the true softmax distribution.”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “Wasserstein distances have a long history and continue to attract interest from diverse fields in statistics, machine learning, and computer science, in particular image analysis; see for instance the monographs and reviews by Santambrogio (2015), Peyré and Cuturi (2019), and Panaretos and Zemel (20”
Learning Deep Kernels for Non-Parametric Two-Sample Tests (2020, background): “…ble or speciﬁc to various more complex domains (Gretton et al., 2012a; Sz´ekely & Rizzo, 2013; Heller & Heller, 2016; Jitkrittum et al., 2016; Ramdas et al., 2017; Lopez-Paz & Oquab, 2017; Chen & Friedman, 2017; Gao et al., 2018; Ghoshdastidar et al., 2017; Ghoshdastidar & von Luxburg,…”
Simulations approaching data: cortical slow waves in inferred models of the whole hemisphere of mouse (2021, neutral): “The proof of the equivalence of both definitions can be found in [76].”
Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations (2015, methodology): “The Wasserstein distance has already been used in tests for normality [17] and to devise nonparametric homogeneity tests [40].”
Sample Complexity of Sinkhorn Divergences (2018, background): “Additionally, recent papers (Ramdas et al., 2017; Genevay et al., 2018) have pointed out the fact that Sinkhorn divergences are in fact interpolating between OT (when regularization goes to zero) and MMD (when regularization goes to inﬁnity).”
Fréchet analysis of variance for random objects (2017, methodology): “m a semimetric derived from the kernel used to compute the MMD. Connections between kernel methods, energy statistics and Wasserstein distance between the distributions to be compared are explored in Ramdas et al. (2017). Commonly used kernels in the kernel framework, like the Gaussian kernel, might”
Interpolating between Optimal Transport and MMD using Sinkhorn Divergences (2018, background): “Following (Genevay et al., 2018) (see also (Ramdas et al., 2017; Salimans et al., 2018; Sanjabi et al., 2018)) we consider a new cost built from OT ε that we call a Sinkhorn divergence : (3) Such a formula satisﬁes S ε ( β, β ) = 0 and interpolates between OT and MMD (Ramdas et al., 2017): The…”
Computational Optimal Transport (2018, background): “The following proposition, whose proof can be found in [Ramdas et al., 2017], shows that this regularized divergence interpolates between the Wasserstein distance and the energy distance defined in Example 8.”
Revisiting Reverse Distillation for Anomaly Detection (2023, methodology): “To achieve this goal, we propose using the de-biased Sinkhorn divergence, a variant of the Optimal Transport distance [7,10,11,26,32].”
CTAB-GAN: Effective Table Data Synthesizing (2021, neutral): “Wasserstein distance (WD) [20].”
Learning Generative Models with Sinkhorn Divergences (2017, methodology): “This theorem is a generalization of [27, §3.3] for continuous measures.”