Convergence analysis of projected fixed-point iteration on a low-rank matrix manifold

TL;DR

This paper analyzes the convergence of projected fixed-point iteration on low-rank matrix manifolds, demonstrating that the projector splitting scheme converges at least as fast as standard fixed-point iteration, supported by theoretical proofs and numerical experiments.

Contribution

It provides a rigorous convergence analysis of the projector splitting scheme on low-rank matrix manifolds, including conditions for convergence and counterexamples.

Findings

Convergence rate matches standard fixed-point iteration under certain conditions

Counterexamples show failure cases when conditions are not met

Numerical experiments validate theoretical results

Abstract

In this paper we analyse convergence of projected fixed-point iteration on a Riemannian manifold of matrices with fixed rank. As a retraction method we use `projector splitting scheme'. We prove that the projector splitting scheme converges at least with the same rate as standard fixed-point iteration without rank constraints. We also provide counter-example to the case when conditions of the theorem do not hold. Finally we support our theoretical results with numerical experiments.