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

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.

Lie groups and continuum mechanics: where do we stand today?

“…The use of classical (vector space) solvers to integrate the equation of these systems may lead to approximate solutions which does not lie on the Lie group and may result in numerical issues. Algorithms that preserve the Lie group structure have been proposed and generally have better stability and robustness properties [100,[148][149][150][151][152][153][154][155]. One of them consists in formulating and solving the differential equation of the system in the corresponding Lie algebra.…”