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 factorized optimization problem. The goal of this problem is to
provide a general theoretical framework for understanding when such methods
work well, and to characterize the nature of the resulting fixed point. We
provide a simple set of conditions under which projected gradient descent, when
given a suitable initialization, converges geometrically to a statistically
useful 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. Working in a non-asymptotic framework, we show that our conditions are
satisfied for a wide range of concrete models, including matrix regression,
structured PCA, matrix completion with real and quantized observations, matrix
decomposition, and graph clustering problems. Simulation results show excellent
agreement with the theoretical predictions.

通过矩阵分解和投影梯度下降算法解决约束最优化问题，提供了一种通用理论框架，当给定适当的初始化时，可以几何级数地收敛到具有统计意义的解，适用于许多具体模型。