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In many applications, one encounters signals that lie on manifolds ratherthan a Euclidean space. In particular, covariance matrices are examples ofubiquitous mathematical objects that have a non Euclidean structure. Theapplication of Euclidean methods to integrate differential equations lying onsuch objects does not respect the geometry of the manifold, which can causemany numerical issues. In this paper, we propose to use Lie group methods todefine geometry-preserving numerical integration schemes on the manifold ofsymmetric positive definite matrices. These can be applied to a number ofdifferential equations on covariance matrices of practical interest. We showthat they are more stable and robust than other classical or naive integrationschemes on an example.

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