Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds

This paper presents two new metrics on the Symmetric Positive Definite (SPD) manifold via the Cholesky manifold, i.e., the space of lower triangular matrices with positive diagonal elements. We first unveil that the existing popular Riemannian metric on the Cholesky manifold can be generally characterized as the product metric of a Euclidean metric and a Riemannian metric on the space of n-dimensional positive vectors. Based on this analysis, we propose two novel metrics on the Cholesky manifolds, i.e., Diagonal Power Euclidean Metric and Diagonal Generalized Bures-Wasserstein Metric, which are numerically stabler than the existing Cholesky metric. We also discuss the gyro structures and deformed metrics associated with our metrics. The gyro structures connect the linear and geometric properties, while the deformed metrics interpolate between our proposed metrics and the existing metric. Further, by Cholesky decomposition, the proposed deformed metrics and gyro structures are pulled back to SPD manifolds. Compared with existing Riemannian metrics on SPD manifolds, our metrics are easy to use, computationally efficient, and numerically stable.