General Low-rank Matrix Optimization: Geometric Analysis and Sharper Bounds

This paper considers the global geometry of general low-rank minimization problems via the Burer-Monterio factorization approach. For the rank-1 case, we prove that there is no spurious second-order critical point for both symmetric and asymmetric problems if the rank-2 RIP constant δ is less than 1/2. Combining with a counterexample with δ=1/2, we show that the derived bound is the sharpest possible. For the arbitrary rank-r case, the same property is established when the rank-2r RIP constant δ is at most 1/3. We design a counterexample to show that the non-existence of spurious second-order critical points may not hold if δ is at least 1/2. In addition, for any problem with δ between 1/3 and 1/2, we prove that all second-order critical points have a positive correlation to the ground truth. Finally, the strict saddle property, which can lead to the polynomial-time global convergence of various algorithms, is established for both the symmetric and asymmetric problems when the rank-2r RIP constant δ is less than 1/3. The results of this paper significantly extend several existing bounds in the literature.