A note on rank constrained solutions to linear matrix equations

TL;DR

This paper introduces a heuristic method called Low-Rank-Functional for finding rank-constrained solutions to linear matrix equations, demonstrating promising simulation results despite lacking formal guarantees.

Contribution

It proposes a novel heuristic based on minimizing a non-convex quadratic functional for rank-constrained LMEs, with empirical validation through numerical examples.

Findings

Performs well in simulations

Offers an intuitive approach

Lacks formal convergence proof

Abstract

This preliminary note presents a heuristic for determining rank constrained solutions to linear matrix equations (LME). The method proposed here is based on minimizing a non-convex quadratic functional, which will hence-forth be termed as the \textit{Low-Rank-Functional} (LRF). Although this method lacks a formal proof/comprehensive analysis, for example in terms of a probabilistic guarantee for converging to a solution, the proposed idea is intuitive and has been seen to perform well in simulations. To that end, many numerical examples are provided to corroborate the idea.