Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing

By Aaditya Ramdas, Sashank J. Reddi, Barnab'as P'oczos, Aarti Singh, Larry A. Wasserman. 2015

48 citations recorded by the source corpus.

Nonparametric two sample testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. We refer to the most common settings as mean difference alternatives (MDA), for testing differences only in first moments, and general difference alternatives (GDA), which is about testing for any difference in distributions. A large number of test statistics have been proposed for both these settings. This paper connects three classes of statistics - high dimensional variants of Hotelling's t-test, statistics based on Reproducing Kernel Hilbert Spaces, and energy statistics based on pairwise distances. We ask the question: how much statistical power do popular kernel and distance based tests for GDA have when the unknown distributions differ in their means, compared to specialized tests for MDA? We formally characterize the power of popular tests for GDA like the Maximum Mean Discrepancy with the Gaussian kernel (gMMD) and bandwidth-dependent variants of the Energy Distance with the Euclidean norm (eED) in the high-dimensional MDA regime. Some practically important properties include (a) eED and gMMD have asymptotically equal power; furthermore they enjoy a free lunch because, while they are additionally consistent for GDA, they also have the same power as specialized high-dimensional t-test variants for MDA. All these tests are asymptotically optimal (including matching constants) under MDA for spherical covariances, according to simple lower bounds, (b) The power of gMMD is independent of the kernel bandwidth, as long as it is larger than the choice made by the median heuristic, (c) There is a clear and smooth computation-statistics tradeoff for linear-time, subquadratic-time and quadratic-time versions of these tests, with more computation resulting in higher power.

Subfields: Computer Networks and Communications, Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Ramdas et al. (2015a) also demonstrates that the power of the kernel-based tests degrades with dimension.”
Revisiting Classifier Two-Sample Tests (2016, background): “These are problems with ﬁxed d and n → ∞ , where the power bounds do not depend on d .”
Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy (2016, methodology): “(2014); Ramdas et al. (2015) study the power of the median heuristic in high-dimensional problems, and justify its use for the case where the means of P and Q differ. An early heuristic for improving test power was to simply maximize M̂MD 2 U. Sriperumbudur et al. (2009) proved that, for certain cla”