This work considers two popular minimization problems: (i) the minimization of a general convex function $f(mathbf{X})$ with the domain being positive semi-definite matrices; (ii) the minimization of a general convex function $f(mathbf{X})$ regularized by the matrix nuclear norm $|mathbf{X}|_*$ with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer and Monteiro to factor the low-rank variable $mathbf{X} = mathbf{U}mathbf{U}^top $ (for semi-definite matrices) or $mathbf{X}=mathbf{U}mathbf{V}^top $ (for general matrices) and also replace the nuclear norm $|mathbf{X}|_*$ with $(|mathbf{U}|_F^2+|mathbf{V}|_F^2)/2$. In spite of the non-convexity of the resulting factored formulations, we prove that each critical point either corresponds to the global optimum of the original convex problems or is a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a nice geometric structure of the factored formulations allows many local search algorithms to find a global optimizer even with random initializations.
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