Symmetry, Saddle Points, and Global Optimization Landscape of Nonconvex Matrix Factorization

We propose a general theory for studying the xllandscape of nonconvex xloptimization with underlying symmetric structures tzfor a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks). In specific, we characterize the locations of stationary points and the null space of Hessian matrices xlof the objective function via the lens of invariant groupsremovedfor associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks. As a major motivating example, we apply the proposed general theory to characterize the global xllandscape of the xlnonconvex optimization in low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: (cR1) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; (cR2) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and (cR3) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. Such global landscape implies strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.