$\ell_1$-norm rank-one symmetric matrix factorization has no spurious second-order stationary points

TL;DR

This paper analyzes the nonsmooth landscape of $\,\ell_1$-norm rank-one symmetric matrix factorization, showing all second-order stationary points are globally optimal, and characterizes the stationary points set.

Contribution

It provides a complete characterization of stationary points and proves the absence of spurious second-order stationary points in the problem.

Findings

All second-order stationary points are globally optimal.

Complete characterization of stationary points set.

Potential application to other nonsmooth learning problems.

Abstract

This paper studies the nonsmooth optimization landscape of the $\ell_1$-norm rank-one symmetric matrix factorization problem using tools from second-order variational analysis. Specifically, as the main finding of this paper, we show that any second-order stationary point (and thus local minimizer) of the problem is actually globally optimal. Besides, some other results concerning the landscape of the problem, such as a complete characterization of the set of stationary points, are also developed, which should be interesting in their own rights. Furthermore, with the above theories, we revisit existing results on the generic minimizing behavior of simple algorithms for nonsmooth optimization and showcase the potential risk of their applications to our problem through several examples. Our techniques can potentially be applied to analyze the optimization landscapes of a variety of other more sophisticated nonsmooth learning problems, such as robust low-rank matrix recovery.