Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition

TL;DR

This paper introduces a new Riemannian metric on SPD matrices using Cholesky decomposition, offering computational simplicity, numerical stability, and a swelling effect-free property, with applications in averaging SPD matrices.

Contribution

The paper develops a Log-Cholesky metric on SPD matrices via a Lie group structure on Cholesky space, improving computational efficiency and stability over existing metrics.

Findings

The Log-Cholesky average preserves determinant bounds, avoiding swelling effects.

The new metric is simpler, faster, and more stable than affine-invariant and Log-Euclidean metrics.

Closed-form parallel transport under the new metric facilitates practical computations.

Abstract

We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Fr\'echet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form.