Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs

Abstract

Collections of probability distributions arise in a variety of statistical applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions are represented by histograms over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of histograms over such general domains. To this end, we introduce the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, which allows us to reduce the histogram collection comparison problem to the comparison of means in a high-dimensional Euclidean space. We develop a hypothesis testing procedure based on conducting t-tests on each dimension of this embedding, then combining the resulting p-values using recently proposed p-value combination techniques. Our numerical experiments in a variety of data settings show that the resulting tests are powerful and the p-values are well-calibrated. Example applications to user activity patterns, spatial data, and brain connectomics are provided.

The Lévy combination test

“…Stable distributions vary in their suitability for the important practical applications of combined testing. Distributions with left and right heavy tails, including the CCT [13,15], exhibit sensitivity to p-values near 1 which is a serious limitation in many practical scenarios. The class of extremal Stable distributions (skewness parameter |β| = 1) that have just one heavy tail therefore appear more suitable.…”

“…Insensitivity to p-values near 1. Despite its advantages over the HMP procedure in terms of its convenient formula and exactness for any number of constituent p-values, the CCT suffers the drawback of undesirable sensitivity to p-values at or near 1 [12,13,15]. Unfortunately this is probably a fatal flaw for the elegant CCT in many settings because generally p-values are defined conservatively such that Pr(p ≤ α|H 0 ) ≤ α; they are said to be 'superuniform' [16].…”

“…Stable distributions is they exhibit robustness to dependence between the constituent p-values despite requiring no knowledge of the explicit dependence structure and despite relying on an independence assumption in their derivation [1,2,10,11,13,15,22,23].…”

Evaluation of First Treatment Timing, Fatal Disease Onset, and Days from First Treatment to Death Associated with Bovine Respiratory Disease in Feedlot Cattle

“…Wasserstein distance (WD) is a calculated metric useful for comparing two distributions and the resulting value is equal to the average distance between two corresponding distributions [ 31 ]. The unit of analysis is the full distribution and the larger WD, the greater the difference between the two distributions [ 32 , 33 , 34 ]. Wasserstein distance is equal to 0 when two distributions perfectly overlap and is greater when distributions are farther apart.…”

Sliced optimal transport on the sphere

“…There exists sliced variants [8,16] of partial optimal transport [19,27], where only a fraction of mass is transported, and a sliced version [20] of multi-marginal optimal transport [10,12,30], considering the transport between several measures instead of only two. For optimal transport on Riemannian manifolds, sliced Wasserstein distances based on the push-forward of the eigenfunctions of the Laplacian have been proposed in [77]. Especially for shape and graph analysis, sliced optimal transport has been transferred to the Gromov-Wasserstein setting [86], which more generally defines a metric between metric measure spaces [11,55,83].…”