  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.
