Flatness of the nuclear norm sphere, simultaneous polarization, and uniqueness in nuclear norm minimization

TL;DR

This paper characterizes the geometric structure of the nuclear norm sphere, introduces conditions for solution uniqueness in nuclear norm minimization problems, and connects these results to broader regularized optimization scenarios.

Contribution

It provides necessary and sufficient conditions for the existence of flats in the nuclear norm sphere and for the uniqueness of solutions in nuclear norm minimization, using novel polarization and subdifferential analysis.

Findings

Characterization of flats in the nuclear norm sphere.

Necessary and sufficient conditions for solution uniqueness.

Extension of results to regularized minimization problems.

Abstract

In this paper we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqueness of solutions for minimizing the nuclear norm over an affine manifold. We further establish an alternative set of sufficient conditions for uniqueness, based on the interplay of the subdifferential of the nuclear norm and the range of the problem-defining linear operator. Finally, using convex duality, we show how to transfer the uniqueness results for the original problem to a whole class of nuclear norm-regularized minimization problems with a strictly convex fidelity term.