Symmetry, Saddle Points, and Global Geometry of Nonconvex Matrix Factorization

• Activity Number: 519 - Sparse Statistical Learning
• Type: Contributed
• Date/Time: Wednesday, August 2, 2017 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Xingguo Li* and Jarvis Haupt and Zhaoran Wang and Junwei Lu and Han Liu and Raman Arora and Tuo Zhao
• Companies: University of Minnesota and University of Minnesota and Princeton University and Princeton University and Princeton University and Johns Hopkins University and Georgia Institute of Technology
• Keywords: Nonconvex Optimization ; Matrix Factorization ; Global Geometry ; Symmetry Group ; Matrix Sensing

Abstract

We propose a general theory for studying the geometry of nonconvex objectives with underlying symmetric structures. In specific, we characterize the locations of stationary points and the null space of the associated Hessian via the lens of invariant groups. As a motivating example, we apply the proposed theory to characterize the global geometry of the low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinite nonisolated strict saddle points and equivalent global minima. By identifying all stationary points, we divide the entire parameter space into three regions: (R1) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; (R2) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and (R3) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. This allows us to establish strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.