Computationally Efficient and Statistically Optimal Robust Low-rank Matrix and Tensor Estimation

TL;DR

This paper introduces RsGrad, a Riemannian sub-gradient algorithm that achieves both computational efficiency and statistical optimality in robust low-rank matrix and tensor estimation under heavy-tailed noise, with a unique dual-phase convergence behavior.

Contribution

The paper proposes a novel RsGrad algorithm that is both computationally efficient and statistically optimal, with a dual-phase convergence phenomenon, applicable to matrices and tensors under heavy-tailed noise.

Findings

RsGrad converges linearly in phase two, achieving statistical optimality.

Dual-phase convergence reveals a smoothing effect of noise on non-smooth losses.

Numerical simulations confirm theoretical advantages and outperform prior methods.

Abstract

Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.