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Implicit Regularization of Sub-Gradient Method in Robust Matrix Recovery: Don't be Afraid of Outliers

Jianhao Ma(jianhao***at***umich.edu )
Salar Fattahi(fattahi***at***umich.edu)

Abstract: It is well-known that simple short-sighted algorithms, such as gradient descent, generalize well in the over-parameterized learning tasks, due to their implicit regularization. However, it is unknown whether the implicit regularization of these algorithms can be extended to robust learning tasks, where a subset of samples may be grossly corrupted with noise. In this work, we provide a positive answer to this question in the context of robust matrix recovery problem. In particular, we consider the problem of recovering a low-rank matrix from a number of linear measurements, where a subset of measurements are corrupted with large noise. We show that a simple sub-gradient method converges to the true low-rank solution efficiently, when it is applied to the over-parameterized L1-loss function without any explicit regularization or rank constraint. Moreover, by building upon a new notion of restricted isometry property, called sign-RIP, we prove the robustness of the sub-gradient method against outliers in the over-parameterized regime. In particular, we show that, with Gaussian measurements, the sub-gradient method is guaranteed to converge to the true low-rank solution, even if an arbitrary fraction of the measurements are grossly corrupted with noise.

Keywords: Nonconvex Optimization, Low-rank Matrix Recovery, Sub-gradient Method

Category 1: Nonlinear Optimization

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 3: Applications -- Science and Engineering (Statistics )

Citation: University of Michigan, Feb 2021

Download: [PDF]

Entry Submitted: 02/06/2021
Entry Accepted: 02/06/2021
Entry Last Modified: 02/06/2021

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