Guarantees of riemannian optimization for low rank matrix completion

We establish the exact recovery guarantees for a class of Riemannian optimization methods based on the embedded manifold of low rank matrices for matrix completion. Assume $ m $ entries of an $ n\times n $ rank $ r $ matrix are sampled independently and uniformly with replacement. We first show that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided $ \begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} $ where $ C_\kappa $ is a numerical constant depending on the condition number of the measured matrix. Then the sampling complexity is further improved to $ \begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} $ via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.