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

Collections of probability distributions arise in a variety of applications\nranging from user activity pattern analysis to brain connectomics. In practice\nthese distributions can be defined over diverse domain types including finite\nintervals, circles, cylinders, spheres, other manifolds, and graphs. This paper\nintroduces an approach for detecting differences between two collections of\ndistributions over such general domains. To this end, we propose the intrinsic\nslicing construction that yields a novel class of Wasserstein distances on\nmanifolds and graphs. These distances are Hilbert embeddable, allowing us to\nreduce the distribution collection comparison problem to a more familiar mean\ntesting problem in a Hilbert space. We provide two testing procedures one based\non resampling and another on combining p-values from coordinate-wise tests. Our\nexperiments in various synthetic and real data settings show that the resulting\ntests are powerful and the p-values are well-calibrated.\n