Prox-regularity of rank constraint sets and implications for algorithms

TL;DR

This paper analyzes the geometric properties of rank-constrained matrix sets, deriving a new normal cone formula, and demonstrates their prox-regularity to ensure local linear convergence of algorithms like alternating projections and steepest descent.

Contribution

It introduces a new normal cone formula for rank constraint sets and proves their prox-regularity, enabling convergence guarantees for algorithms in nonconvex rank-constrained problems.

Findings

Normal cone formula for rank sets is new.

Rank sets are prox-regular at points with full rank.

Algorithms like alternating projections converge locally linearly.

Abstract

We present an analysis of sets of matrices with rank less than or equal to a specified number $s$. We provide a simple formula for the normal cone to such sets, and use this to show that these sets are prox-regular at all points with rank exactly equal to $s$. The normal cone formula appears to be new. This allows for easy application of prior results guaranteeing local linear convergence of the fundamental alternating projection algorithm between sets, one of which is a rank constraint set. We apply this to show local linear convergence of another fundamental algorithm, approximate steepest descent. Our results apply not only to linear systems with rank constraints, as has been treated extensively in the literature, but also nonconvex systems with rank constraints.