Vector Transport Free Riemannian LBFGS for Optimization on Symmetric Positive Definite Matrix Manifolds

TL;DR

This paper introduces a novel approach to optimize on SPD manifolds by making vector transports isometric and reducing computational costs, enhancing the efficiency of Riemannian LBFGS algorithms.

Contribution

It proposes two new mappings that simplify vector transports, making RLBFGS more efficient and easier to analyze on SPD manifolds.

Findings

Vector transport becomes isometric and computationally cheaper.

Riemannian metric reduces to Euclidean inner product under the new mappings.

Enhanced convergence analysis due to isometry property.

Abstract

This work concentrates on optimization on Riemannian manifolds. The Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a commonly used quasi-Newton method for numerical optimization in Euclidean spaces. Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian manifolds. RLBFGS involves computationally expensive vector transports as well as unfolding recursions using adjoint vector transports. In this article, we propose two mappings in the tangent space using the inverse second root and Cholesky decomposition. These mappings make both vector transport and adjoint vector transport identity and therefore isometric. Identity vector transport makes RLBFGS less computationally expensive and its isometry is also very useful in convergence analysis of RLBFGS. Moreover, under the proposed mappings, the Riemannian metric reduces to Euclidean inner product, which is much less computationally expensive. We focus on the Symmetric Positive Definite (SPD) manifolds which are beneficial in various fields such as data science and statistics. This work opens a research opportunity for extension of the proposed mappings to other well-known manifolds.