Closed-Form Solutions to A Category of Nuclear Norm Minimization Problems

TL;DR

This paper derives a closed-form solution for a specific nuclear norm minimization problem related to low-rank representation, which is significant in machine learning and computer vision applications.

Contribution

It proves a unique, closed-form solution for a class of nuclear norm minimization problems, advancing theoretical understanding in low-rank matrix recovery.

Findings

Established a closed-form solution for the LRR problem

Proved the solution's uniqueness and optimality

Provided a lemma for solving a category of nuclear norm problems

Abstract

It is an efficient and effective strategy to utilize the nuclear norm approximation to learn low-rank matrices, which arise frequently in machine learning and computer vision. So the exploration of nuclear norm minimization problems is gaining much attention recently. In this paper we shall prove that the following Low-Rank Representation (LRR) \cite{icml_2010_lrr,lrr_extention} problem: {eqnarray*} \min_{Z} \norm{Z}_*, & {s.t.,} & X=AZ, {eqnarray*} has a unique and closed-form solution, where $X$ and $A$ are given matrices. The proof is based on proving a lemma that allows us to get closed-form solutions to a category of nuclear norm minimization problems.