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This paper studies the role of over-parametrization in solving non-convexoptimization problems. The focus is on the important class of low-rank matrixsensing, where we propose an infinite hierarchy of non-convex problems via thelifting technique and the Burer-Monteiro factorization. This contrasts with theexisting over-parametrization technique where the search rank is limited by thedimension of the matrix and it does not allow a rich over-parametrization of anarbitrary degree. We show that although the spurious solutions of the problemremain stationary points through the hierarchy, they will be transformed intostrict saddle points (under some technical conditions) and can be escaped vialocal search methods. This is the first result in the literature showing thatover-parametrization creates a negative curvature for escaping spurioussolutions. We also derive a bound on how much over-parametrization is requitedto enable the elimination of spurious solutions.

Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points