Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima

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.