The nonconvex geometry of low-rank matrix optimizations with general objective functions

This work considers the minimization of a general convex function f (X) over the cone of positive semi-definite matrices whose optimal solution X∗ is of low-rank. Standard first-order convex solvers require performing an eigenvalue decomposition in each iteration, severely limiting their scalability. A natural nonconvex reformulation of the problem factors the variable X into the product of a rectangular matrix with fewer columns and its transpose. For a special class of matrix sensing and completion problems with quadratic objective functions, local search algorithms applied to the factored problem have been shown to be much more efficient and, in spite of being nonconvex, to converge to the global optimum. The purpose of this work is to extend this line of study to general convex objective functions f (X) and investigate the geometry of the resulting factored formulations. Specifically, we prove that when f (X) satisfies the restricted well-conditioned assumption, each critical point of the factored problem either corresponds to the optimal solution X∗ or a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a geometric structure of the factored formulation ensures that many local search algorithms can converge to the global optimum with random initializations.