Riemannian Perspective on Matrix Factorization

Abstract

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.

arXiv comment23 pages, 6 figures. Comments would be appreciated!

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Cited by in corpus (2)

• Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization