Optimal tuning-free convex relaxation for noisy matrix completion
Yuepeng Yang, Cong Ma
TL;DR
We study noisy matrix completion under uniform sampling and $mu$-incoherence and introduce a tuning-free square-root matrix completion estimator, square-root MC, whose regularization parameter scales as $\lambda \asymp 1/\sqrt{n}$. The authors establish minimax-optimal Frobenius and entrywise error bounds that depend on the condition number $\kappa$, incoherence $\mu$, rank $r$, sampling probability $p$, and noise level $\sigma$, with high probability. A key methodological contribution is linking the convex square-root MC to a smooth nonconvex reformulation via a new variable $\theta$, and proving that an approximate stationary point of the nonconvex problem lies close to the ground truth and to the convex solution; leave-one-out techniques are used to control the iterates. Overall, the tuning-free estimator achieves optimal statistical performance without knowledge of the noise size, offering practical advantages and providing a blueprint for connecting convex and nonconvex approaches in high-dimensional recovery.
Abstract
This paper is concerned with noisy matrix completion--the problem of recovering a low-rank matrix from partial and noisy entries. Under uniform sampling and incoherence assumptions, we prove that a tuning-free square-root matrix completion estimator (square-root MC) achieves optimal statistical performance for solving the noisy matrix completion problem. Similar to the square-root Lasso estimator in high-dimensional linear regression, square-root MC does not rely on the knowledge of the size of the noise. While solving square-root MC is a convex program, our statistical analysis of square-root MC hinges on its intimate connections to a nonconvex rank-constrained estimator.