A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix Estimation
This provides a unified framework for low-rank matrix estimation, applicable to problems like matrix completion, but it is incremental as it builds on existing nonconvex methods.
The authors tackled the problem of estimating low-rank matrices using nonconvex optimization, achieving linear convergence to minimax optimal statistical error in noisy settings and exact recovery with optimal sample complexity in noiseless settings.
We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general case with noisy observations, we show that our algorithm is guaranteed to linearly converge to the unknown low-rank matrix up to minimax optimal statistical error, provided an appropriate initial estimator. While in the generic noiseless setting, our algorithm converges to the unknown low-rank matrix at a linear rate and enables exact recovery with optimal sample complexity. In addition, we develop a new initialization algorithm to provide a desired initial estimator, which outperforms existing initialization algorithms for nonconvex low-rank matrix estimation. We illustrate the superiority of our framework through three examples: matrix regression, matrix completion, and one-bit matrix completion. We also corroborate our theory through extensive experiments on synthetic data.