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.
Highlights
Rank optimization is a well-developed topic that has found a tremendous number of applications in recent years
Prox-regularity of the lower level sets of the rank function immediately yields local linear convergence of fundamental algorithms for either finding the intersection of the rank constraint set with another set determined by some data model, or for minimizing the distance to a rank constrained set and a data set
We have developed a novel characterization of the normal cone to the lower level sets of the rank function
Summary
Rank optimization is a well-developed topic that has found a tremendous number of applications in recent years (see [23] and references therein). Prox-regularity of the lower level sets of the rank function immediately yields local linear convergence of fundamental algorithms for either finding the intersection of the rank constraint set with another set determined by some (nonlinear) data model, or for minimizing the distance to a rank constrained set and a data set. In any case, avoiding convex surrogates is at the cost of global convergence guarantees: these results are local and offer no panacea for solving rank optimization problems. Rather, this analysis shows that certain macro-regularity assumptions such as restricted isometry or mutual coherence (see [23] and references therein) play no role asymptotically in the convergence of algorithms, but rather have bearing only on the radius of convergence. We begin this note with a review of notation and basic results and definitions upon which we build
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