Many fundamental low-rank optimization problems, such as matrix completion, phase retrieval, and robust PCA, can be formulated as the matrix sensing problem. Two main approaches for solving matrix sensing are based on semidefinite programming (SDP) and Burer-Monteiro (B-M) factorization. The former suffers from high computational and space complexities, whereas the latter may return a spurious solution due to the non-convexity of the problem. The existing theoretical guarantees for the success of these methods have led to similar conservative conditions, which may wrongly imply that these methods have comparable performances. In this paper, we shed light on some major differences between these two methods. First, we present a class of structured matrix completion problems for which the B-M methods fail with an overwhelming probability, while the SDP method works correctly. Second, we identify a class of highly sparse matrix completion problems for which the B-M method works and the SDP method fails. Third, we prove that although the B-M method exhibits the same performance independent of the rank of the unknown solution, the success of the SDP method is correlated to the rank of the solution and improves as the rank increases. Unlike the existing literature that has mainly focused on those instances of matrix sensing for which both SDP and B-M work, this paper offers the first result on the unique merit of each method over the alternative approach.

Burer-Monteiro (B-M) factorization approach can efficiently solve low-rank matrix optimization problems under the Restricted Isometry Property (RIP) condition. It is natural to ask whether B-M factorization-based methods can succeed on any low-rank matrix optimization problems with low information-theoretic complexity, i.e., polynomial-time solvable problems that have a unique solution. We provide negative answer to this question. We investigate the landscape of B-M factorized polynomial-time solvable matrix completion (MC) problems, which are the most popular subclass of low-rank matrix optimization problems without the RIP condition. We construct an instance of polynomial-time solvable MC problems with exponentially many spurious local minima, which leads to the failure of most gradient-based methods. We define a new complexity metric that measures the solvability of low-rank matrix optimization problems based on B-M factorization approach. In addition, we show that more measurements can deteriorate the landscape, which further reveals the unfavorable behavior of B-M factorization.