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We study the problem of estimating low-rank matrices from linear measurements(a.k.a., matrix sensing) through nonconvex optimization. We propose anefficient stochastic variance reduced gradient descent algorithm to solve anonconvex optimization problem of matrix sensing. Our algorithm is applicableto both noisy and noiseless settings. In the case with noisy observations, weprove that our algorithm converges to the unknown low-rank matrix at a linearrate up to the minimax optimal statistical error. And in the noiseless setting,our algorithm is guaranteed to linearly converge to the unknown low-rank matrixand achieves exact recovery with optimal sample complexity. Most notably, theoverall computational complexity of our proposed algorithm, which is defined asthe iteration complexity times per iteration time complexity, is lower than thestate-of-the-art algorithms based on gradient descent. Experiments on syntheticdata corroborate the superiority of the proposed algorithm over thestate-of-the-art algorithms.

Stochastic Variance-reduced Gradient Descent for Low-rank Matrix Recovery from Linear Measurements