High-order Accurate Inference on Manifolds

TL;DR

This paper introduces a high-order accurate statistical inference framework on Riemannian manifolds, enabling precise hypothesis testing and confidence regions by incorporating manifold geometry and curvature effects.

Contribution

It develops a novel, computationally efficient bootstrap algorithm for high-order inference on manifolds, extending traditional Euclidean methods to complex geometric spaces.

Findings

The framework achieves high-order asymptotic accuracy in manifold inference.

Numerical studies demonstrate improved precision over existing methods.

Applicability across various manifolds including spheres and tensor manifolds.

Abstract

We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under an appropriate coordinate representation induced by a second-order retraction, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings, including spheres, the Stiefel manifold, fixed-rank matrix manifolds, and rank-one tensor manifolds; for Euclidean submanifolds, we also introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.