Riemannian Perspective on Matrix Factorization
This work provides theoretical insights into the optimization landscape for matrix completion, which is an incremental contribution to the field of machine learning and optimization.
The authors tackled the non-convex optimization problem in matrix factorization for matrix completion by analyzing it through Riemannian geometry on a Grassmannian manifold, showing that in the fully observed case, there exists a region of geodesic convexity and that all critical points outside this region are strictly saddle points.
We study the non-convex matrix factorization approach to matrix completion via Riemannian geometry. Based on an optimization formulation over a Grassmannian manifold, we characterize the landscape based on the notion of principal angles between subspaces. For the fully observed case, our results show that there is a region in which the cost is geodesically convex, and outside of which all critical points are strictly saddle. We empirically study the partially observed case based on our findings.