The Non-convex Geometry of Low-rank Matrix Optimization

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.