Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima

TL;DR

This paper investigates the nonconvex landscape of nonnegative low-rank matrix recovery, revealing that benign geometry observed in ideal cases does not generally extend to more realistic scenarios, challenging existing theoretical explanations.

Contribution

It demonstrates that benign nonconvexity in nonnegative low-rank recovery is limited to idealized cases and does not hold under partial observations or higher ranks, highlighting the need for new analytical tools.

Findings

Benign nonconvexity holds for rank-1, fully observed case with RIP=0.

Benign nonconvexity fails under partial observations with any RIP > 0.

Higher-rank cases also lack benign geometry, regardless of overparameterization.

Abstract

Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $\delta=0$. This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant $\delta>0$, and to higher-rank ground truths $r^{\star}>1$, regardless of how much the search rank $r\ge r^{\star}$ is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery.