A novel representation of rank constraints for non-square real matrices

TL;DR

This paper introduces a new way to represent rank constraints for non-square real matrices, linking it to existing results and applications like sparse representation and rank minimization.

Contribution

It proposes a novel mathematical representation of rank constraints that generalizes previous results and can be integrated into optimization problems.

Findings

Unified framework for rank constraints and $\, ext{ extlangle}0 ext{ extrangle}$ pseudo-norm

Connections established with existing rank representations

Potential applications in rank-constrained optimization

Abstract

We present a novel representation of rank constraints for non-square real matrices. We establish relationships with some existing results, which are particular cases of our representation. One of these particular cases, is a representation of the $\ell_0$ pseudo-norm, which is used in sparse representation problems. Finally, we describe how our representation can be included in rank-constrained optimization and in rank-minimization problems.