Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
This paper establishes that the Burer-Monteiro factorization of fixed-rank positive semidefinite matrix optimization problems exhibits a favorable global landscape under Riemannian quotient geometry, dividing the search space into regions of geodesic strong convexity, strict saddle neighborhoods, and large gradients, thereby providing a geometric explanation for the success of vanilla gradient descent.