Abstract:
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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.
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