Riemannian Stein Variational Gradient Descent for Bayesian Inference

TL;DR

This paper introduces Riemannian Stein Variational Gradient Descent (RSVGD), a novel Bayesian inference method that extends SVGD to Riemann manifolds, leveraging information geometry for improved inference in Euclidean and Riemannian spaces.

Contribution

The paper develops RSVGD with new techniques for Riemann manifolds, introduces Riemannian Stein's Identity and Kernelized Stein Discrepancy, and demonstrates its advantages over existing methods.

Findings

RSVGD outperforms SVGD by utilizing distribution geometry.

RSVGD shows particle-efficiency and iteration-effectiveness.

Experimental results confirm advantages on Riemann manifolds.

Abstract

We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and (ii) for inference tasks on Riemann manifolds, RSVGD brings the unique advantages of SVGD to the Riemannian world. To appropriately transfer to Riemann manifolds, we conceive novel and non-trivial techniques for RSVGD, which are required by the intrinsically different characteristics of general Riemann manifolds from Euclidean spaces. We also discover Riemannian Stein's Identity and Riemannian Kernelized Stein Discrepancy. Experimental results show the advantages over SVGD of exploring distribution geometry and the advantages of particle-efficiency, iteration-effectiveness and approximation flexibility over other inference methods on Riemann manifolds.