Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition

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 Fréchet 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.

arXiv comment19 pages, 1 figure

References in corpus (1)

• Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging

Cited by in corpus (15)

• Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies
• QGOpt: Riemannian optimization for quantum technologies
• Deep Efficient Continuous Manifold Learning for Time Series Modeling
• A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions
• The Comparison of Riemannian Geometric Matrix-CFAR Signal Detectors
• Topology of surface electromyogram signals: hand gesture decoding on Riemannian manifolds
• On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry
• Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups
• Exponential Concentration for Geometric-Median-of-Means in Non-Positive Curvature Spaces
• Dynamical Wasserstein Barycenters for Time-series Modeling
• The exponentially weighted average forecaster in geodesic spaces of non-positive curvature
• Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry
• GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition
• Barycenter Estimation of Positive Semi-Definite Matrices with Bures-Wasserstein Distance
• Wasserstein Regression