On the Absence of Spurious Local Minima in Nonlinear Low-Rank Matrix Recovery Problems

The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, for general low-rank matrix recovery problems with nonlinear measurements, a novel property named bound difference property (BDP) is introduced. Using RIP and BDP jointly, we propose a new criterion to certify the nonexistence of spurious local minima in the rank-1 case, and prove that it leads to a much stronger theoretical guarantee than the existing bounds on RIP.