A Bayesian Approach for Analyzing Data on the Stiefel Manifold

TL;DR

This paper introduces a Bayesian framework for analyzing directional data on the Stiefel manifold, utilizing the Matrix Langevin distribution with new conjugate priors and a novel computational method for posterior inference.

Contribution

It develops a unified Bayesian approach with conjugate priors for the Stiefel manifold and proposes a new computational technique for hypergeometric functions of matrix arguments.

Findings

Established theoretical properties of the priors and posteriors.

Developed a novel computational procedure for hypergeometric functions.

Enabled efficient posterior sampling for directional data analysis.

Abstract

Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference. %Importantly, these include the propriety of these priors and concentration characterization. Conjugacy enables the translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.