Low rank matrix recovery with adversarial sparse noise*This work was supported in part by National Natural Science Foundation of China under Grant numbers 12071426, 11971427, 11901518, the NSAF of China under grant number U21A20426, and the Fundamental Research Funds for the Central Universities under Grant number 2020XZZX002-03. (18th January 2022) - Xu, Hang; Li, Song; Lin, Junhong
Abstract: Many problems in data science can be treated as recovering a low-rank matrix from a small number of random linear measurements, possibly corrupted with adversarial noise and dense noise. Recently, a bunch of theories on variants of models have been developed for different noises, but with fewer theories on the adversarial noise. In this paper, we study low-rank matrix recovery problem from linear measurements perturbed by ℓ 1 -bounded noise and sparse noise that can arbitrarily change an adversarially chosen ω -fraction of the measurement vector. For Gaussian measurements with nearly optimal number of measurements, we show that the nuclear-norm constrained least absolute deviation (LAD) can successfully estimate the ground-truth matrix for any ω < 0.239. Similar robust recovery results are also established for an iterative hard thresholding algorithm applied to the rank-constrained LAD considering geometrically decaying step-sizes, and the unconstrained LAD based on matrix factorization as well as its subgradient descent solver.