Geometry-preserving Lie Group Integrators For Differential Equations On The Manifold Of Symmetric Positive Definite Matrices

TL;DR

This paper introduces Lie group integrators that preserve the geometry of the manifold of symmetric positive definite matrices, improving stability and robustness in numerical solutions of differential equations on covariance matrices.

Contribution

It proposes a novel class of geometry-preserving Lie group integrators specifically designed for differential equations on the manifold of symmetric positive definite matrices.

Findings

More stable than classical schemes

Robustness in practical applications

Effective for covariance matrix differential equations

Abstract

In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.