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Optimization problems with rank constraints arise in many applications,including matrix regression, structured PCA, matrix completion and matrixdecomposition problems. An attractive heuristic for solving such problems is tofactorize the low-rank matrix, and to run projected gradient descent on thenonconvex factorized optimization problem. The goal of this problem is toprovide a general theoretical framework for understanding when such methodswork well, and to characterize the nature of the resulting fixed point. Weprovide a simple set of conditions under which projected gradient descent, whengiven a suitable initialization, converges geometrically to a statisticallyuseful solution. Our results are applicable even when the initial solution isoutside any region of local convexity, and even when the problem is globallyconcave. Working in a non-asymptotic framework, we show that our conditions aresatisfied for a wide range of concrete models, including matrix regression,structured PCA, matrix completion with real and quantized observations, matrixdecomposition, and graph clustering problems. Simulation results show excellentagreement with the theoretical predictions.

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