Bayesian Geodesic Regression on Riemannian Manifolds

TL;DR

This paper introduces a Bayesian geodesic regression model on Riemannian manifolds that automatically determines data dimensionality and reduces overfitting, demonstrated on synthetic and real shape data.

Contribution

It develops a novel Bayesian geodesic regression framework with automatic dimensionality selection and regularization for Riemannian manifold data.

Findings

Effective in reducing data dimensionality

Successfully applied to shape variation analysis

Demonstrates improved model robustness

Abstract

Geodesic regression has been proposed for fitting the geodesic curve. However, it cannot automatically choose the dimensionality of data. In this paper, we develop a Bayesian geodesic regression model on Riemannian manifolds (BGRM) model. To avoid the overfitting problem, we add a regularization term to control the effectiveness of the model. To automatically select the dimensionality, we develop a prior for the geodesic regression model, which can automatically select the number of relevant dimensions by driving unnecessary tangent vectors to zero. To show the validation of our model, we first apply it in the 3D synthetic sphere and 2D pentagon data. We then demonstrate the effectiveness of our model in reducing the dimensionality and analyzing shape variations of human corpus callosum and mandible data.