On the Convex Geometry of Weighted Nuclear Norm Minimization

TL;DR

This paper analyzes the convex geometry of weighted nuclear norm minimization (WNNM), showing it is convex with a unique solution when weights are non-descending, thus providing theoretical insights into low-rank matrix approximation.

Contribution

It establishes the convexity and uniqueness of the WNNM solution under specific weight conditions, connecting convex geometry with matrix recovery algorithms.

Findings

WNNM is convex with non-descending weights.

Unique global minimizer exists for WNNM.

The analysis links convex geometry to compressed sensing.

Abstract

Low-rank matrix approximation, which aims to construct a low-rank matrix from an observation, has received much attention recently. An efficient method to solve this problem is to convert the problem of rank minimization into a nuclear norm minimization problem. However, soft-thresholding of singular values leads to the elimination of important information about the sensed matrix. Weighted nuclear norm minimization (WNNM) has been proposed, where the singular values are assigned different weights, in order to treat singular values differently. In this paper the solution for WNNM is analyzed under a particular weighting condition using the connection between convex geometry and compressed sensing algorithms. It is shown that the WNNM is convex where the weights are in non-descending order and there is a unique global minimizer for the minimization problem.