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Optimization over low rank matrices has broad applications in machinelearning. For large scale problems, an attractive heuristic is to factorize thelow rank matrix to a product of two much smaller matrices. In this paper, westudy the nonconvex problem $\min_{U\in\mathcal{R}^{n\times r}} g(U)=f(UU^T)$under the assumptions that $f(X)$ is restricted $\mu$-strongly convex and$L$-smooth on the set $\{X:X\succeq 0,rank(X)\leq r\}$. We propose anaccelerated gradient method with alternating constraint that operates directlyon the $U$ factors and show that the method has local linear convergence ratewith the optimal dependence on the condition number of $\sqrt{L/\mu}$.Globally, our method converges to the critical point with zero gradient fromany initializer. Our method also applies to the problem with the asymmetricfactorization of $X=\widetilde U\widetilde V^T$ and the same convergence resultcan be obtained. Extensive experimental results verify the advantage of ourmethod.

Provable Accelerated Gradient Method for Nonconvex Low Rank Optimization