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

Raif M. Rustamov, Subhabrata Majumdar

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. We propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the histogram collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures, one based on resampling and another on combining $p$-values from coordinate-wise tests. Our 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 and spatial data are provided.

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