  It is well-known that the Burer-Monteiro (B-M) factorization approach can
efficiently solve low-rank matrix optimization problems under the RIP
condition. It is natural to ask whether B-M factorization-based methods can
succeed on any low-rank matrix optimization problems with a low
information-theoretic complexity, i.e., polynomial-time solvable problems that
have a unique solution. In this work, we provide a negative answer to the above
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. Based on those results, we define a new complexity
metric that potentially measures the solvability of low-rank matrix
optimization problems based on the B-M factorization approach. In addition, we
show that more measurements of the ground truth matrix can deteriorate the
landscape, which further reveals the unfavorable behavior of the B-M
factorization on general low-rank matrix optimization problems.
