A1557
Title: Detecting non-uniform patterns on high-dimensional hyperspheres
Authors: Tuan Pham - University of Texas, Austin (United States) [presenting]
Abstract: A new probabilistic characterization of the uniform distribution on the hypersphere in terms of the distribution of pairwise inner products is proposed. This characterization leads naturally to an Ingster-type distance for quantifying deviations from uniformity, whose asymptotic behavior can be analyzed systematically via Edgeworth-type expansions. Surprisingly, it is shown that this distance captures the minimax rates for testing uniformity simultaneously across several high-dimensional parametric models, even in the models where densities with respect to the uniform law do not exist. A simple test for spherical uniformity based on this distance is proposed. The detection rates and consistency under various classes of alternatives, both local and non-local, are studied. The proposed test is universally consistent in fixed dimensions, minimax-optimal in a variety of high-dimensional parametric models, and consistent against non-local high-dimensional alternatives. This is different from previously studied high-dimensional Sobolev tests and extreme-value-based tests, which are rate-suboptimal or inconsistent against one or more classes of alternatives.
