Nonsmooth rank-one matrix factorization landscape

TL;DR

This paper analyzes the optimization landscape of nonsmooth rank-one matrix factorization in robust PCA, establishing conditions for the absence of spurious local minima, thus advancing understanding of nonsmooth nonconvex problems.

Contribution

It provides the first positive result on the landscape of nonsmooth robust PCA, identifying necessary and sufficient conditions for no spurious local minima in the rank-one case.

Findings

Characterizes when no spurious local minima exist

Utilizes subdifferential regularity to analyze the landscape

Advances understanding of nonsmooth optimization in robust PCA

Abstract

We provide the first positive result on the nonsmooth optimization landscape of robust principal component analysis, to the best of our knowledge. It is the object of several conjectures and remains mostly uncharted territory. We identify a necessary and sufficient condition for the absence of spurious local minima in the rank-one case. Our proof exploits the subdifferential regularity of the objective function in order to eliminate the existence quantifier from the first-order optimality condition known as Fermat's rule.