Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means

TL;DR

This paper introduces a new Riemannian metric on the cone of positive-definite matrices derived from a power potential, providing explicit geodesics and distances that generalize and relate to known matrix means.

Contribution

It defines a novel metric based on the Hessian of the power potential, explicitly characterizes geodesics and distances, and connects these to classical matrix means as parameters vary.

Findings

Explicit formulas for geodesics and distances under the new metric.

In the scalar case, the geodesic matches a weighted power mean.

For matrices, the new mean differs from existing ones and converges to the matrix geometric mean as eta approaches zero.

Abstract

We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function $(1-\det(X)^\beta)/\beta$. We give explicit expressions for the geodesics and distance function, under suitable conditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As $\beta$ tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively.