Certifying optimality in nonconvex robust PCA

TL;DR

This paper analyzes the nonconvex optimization landscape of robust PCA, showing that under certain conditions, all rank-$r$ factorizations are critical points, with local minima at true solutions and saddle points elsewhere.

Contribution

It provides a theoretical characterization of the critical points and local geometry of the nonconvex robust PCA objective under standard assumptions.

Findings

All rank-$r$ factorizations are Clarke critical points.

True solutions are sharp local minima.

Overparameterized factorizations are strict saddle points.

Abstract

Robust principal component analysis seeks to recover a low-rank matrix from fully observed data with sparse corruptions. A scalable approach fits a low-rank factorization by minimizing the sum of entrywise absolute residuals, leading to a nonsmooth and nonconvex objective. Under standard incoherence conditions and a random model for the corruption support, we study factorizations of the ground-truth rank-$r$ matrix with both factors of rank $r$. With high probability, every such factorization is a Clarke critical point. We also characterize the local geometry: when the factorization rank equals $r$, these solutions are sharp local minima; when it exceeds $r$, they are strict saddle points.