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Fast low-rank estimation by projected gradient descent: Statistical and algorithmic guarantees

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If you have a question about this talk, please contact Quentin Berthet .

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA , matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex problem in the lower dimensional factorized space. We provide a general set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically optimal solution. Our results are applicable even when the initial solution is outside any region of local convexity, and even when the problem is globally concave.

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