A Non-Euclidean Gradient Descent Framework for Non-Convex Matrix Factorization

We study convex optimization problems that feature low-rank matrix solutions. In such scenarios, non-convex methods offer significant advantages over convex methods due to their lower space complexity, as well as practical faster convergence. Under mild assumptions, these methods feature global convergence guarantees. In this paper, we extend the results on this matter by following a different path. We derive a non-Euclidean optimization framework in the non-convex setting that takes nonlinear gradient steps on the factors. Our framework enables the possibility to further exploit the underlying problem structures, such as sparsity or low-rankness on the factorized domain, or better dimensional dependence of the smoothness parameters of the objectives. We prove that the non-Euclidean methods enjoy the same rigorous guarantees as their Euclidean counterparts under appropriate assumptions. Numerical evidence with Fourier ptychography and FastText applications, using real data, shows that our approach can enhance solution quality, as well as convergence speed over the standard non-convex approaches.