Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank

TL;DR

This paper develops a novel Riemannian metric and geometric mean for fixed-rank positive semidefinite matrices, enabling efficient computation and preserving key geometric properties.

Contribution

It introduces a new Riemannian quotient geometry-based metric and mean that generalize existing structures and are computationally efficient.

Findings

The metric is geodesically complete and invariant under key transformations.

The proposed distance approximation can be computed efficiently using SVD.

The geometric mean preserves matrix rank and has desirable properties.

Abstract

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute.