Restricted Riemannian geometry for positive semidefinite matrices

TL;DR

This paper introduces a new Riemannian geometric framework for the manifold of fixed-rank positive semidefinite matrices, enabling analytical solutions and improved algorithms for eigenspace estimation.

Contribution

It defines a dense submanifold with a complete Riemannian structure and Lie group properties, providing closed-form formulas and algorithms for key operations.

Findings

Closed-form expressions for geodesics and parallel transport.

A new algorithm for principal eigenspace estimation.

Superior performance of the proposed eigenspace algorithm.

Abstract

We introduce the manifold of {\it restricted} $n\times n$ positive semidefinite matrices of fixed rank $p$, denoted $S(n,p)^{*}$. The manifold itself is an open and dense submanifold of $S(n,p)$, the manifold of $n\times n$ positive semidefinite matrices of the same rank $p$, when both are viewed as manifolds in $\mathbb{R}^{n\times n}$. This density is the key fact that makes the consideration of $S(n,p)^{*}$ statistically meaningful. We furnish $S(n,p)^{*}$ with a convenient, and geodesically complete, Riemannian geometry, as well as a Lie group structure, that permits analytical closed forms for endpoint geodesics, parallel transports, Fr\'echet means, exponential and logarithmic maps. This task is done partly through utilizing a {\it reduced} Cholesky decomposition, whose algorithm is also provided. We produce a second algorithm from this framework to estimate principal eigenspaces and demonstrate its superior performance over other existing algorithms.