Interpretable Stein Goodness-of-fit Tests on Riemannian Manifolds

TL;DR

This paper introduces new goodness-of-fit tests for data on Riemannian manifolds using kernel Stein discrepancy, enabling model validation for complex manifold distributions.

Contribution

It extends kernel Stein discrepancy methods to Riemannian manifolds, including intractable distributions, with theoretical analysis and practical validation.

Findings

Proposed tests are valid for manifold data.

The methods demonstrate high efficiency in simulations.

Applications show effectiveness on real Riemannian data.

Abstract

In many applications, we encounter data on Riemannian manifolds such as torus and rotation groups. Standard statistical procedures for multivariate data are not applicable to such data. In this study, we develop goodness-of-fit testing and interpretable model criticism methods for general distributions on Riemannian manifolds, including those with an intractable normalization constant. The proposed methods are based on extensions of kernel Stein discrepancy, which are derived from Stein operators on Riemannian manifolds. We discuss the connections between the proposed tests with existing ones and provide a theoretical analysis of their asymptotic Bahadur efficiency. Simulation results and real data applications show the validity of the proposed methods.