Efficient Proximal Mapping Computation for Unitarily Invariant Low-Rank Inducing Norms

TL;DR

This paper introduces an efficient framework for computing proximal mappings of low-rank inducing unitarily invariant norms, enabling improved optimization methods for low-rank matrix problems.

Contribution

It provides a nested binary search approach to evaluate proximal mappings for a broad class of low-rank inducing norms, extending beyond the nuclear norm.

Findings

Framework reduces proximal mapping evaluation to nested binary search.

Analytical solutions demonstrated for Frobenius and spectral norms.

Enables computation of proximal mappings for compositions with convex functions.

Abstract

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. This simpler problem can often be solved analytically as it is demonstrated for the so-called low-rank inducing Frobenius and spectral norms. Moreover, the framework allows to compute the proximal mapping of compositions of these norms with increasing convex functions and the projections onto their epigraphs. This has the additional advantage that we can also deal with compositions of increasing convex functions and low-rank inducing norms in proximal splitting methods.