  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.
