Stochastic Variance-reduced Gradient Descent for Low-rank Matrix Recovery from Linear Measurements
This work addresses matrix sensing, a key problem in machine learning and signal processing, by providing a more efficient algorithm, though it is incremental as it builds on existing gradient descent approaches.
The paper tackles low-rank matrix recovery from linear measurements by proposing a stochastic variance-reduced gradient descent algorithm, which achieves linear convergence to the unknown matrix with optimal statistical error in noisy settings and exact recovery with optimal sample complexity in noiseless settings, while reducing overall computational complexity compared to state-of-the-art methods.
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex optimization problem of matrix sensing. Our algorithm is applicable to both noisy and noiseless settings. In the case with noisy observations, we prove that our algorithm converges to the unknown low-rank matrix at a linear rate up to the minimax optimal statistical error. And in the noiseless setting, our algorithm is guaranteed to linearly converge to the unknown low-rank matrix and achieves exact recovery with optimal sample complexity. Most notably, the overall computational complexity of our proposed algorithm, which is defined as the iteration complexity times per iteration time complexity, is lower than the state-of-the-art algorithms based on gradient descent. Experiments on synthetic data corroborate the superiority of the proposed algorithm over the state-of-the-art algorithms.