  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.
