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

arXiv: https://arxiv.org/abs/2203.00953 · HTML: https://mlsanity.com/paper/2203.00953/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2022.

## Built on by (most cited first, 1 of 1)

- [Online Tensor Learning: Computational and Statistical Trade-offs, Adaptivity and Optimal Regret](https://mlsanity.com/paper/2306.03372/) (2023)

## Abstract (arXiv)

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.