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Recent advances have shown that implicit bias of gradient descent onover-parameterized models enables the recovery of low-rank matrices from linearmeasurements, even with no prior knowledge on the intrinsic rank. In contrast,for robust low-rank matrix recovery from grossly corrupted measurements,over-parameterization leads to overfitting without prior knowledge on both theintrinsic rank and sparsity of corruption. This paper shows that with a doubleover-parameterization for both the low-rank matrix and sparse corruption,gradient descent with discrepant learning rates provably recovers theunderlying matrix even without prior knowledge on neither rank of the matrixnor sparsity of the corruption. We further extend our approach for the robustrecovery of natural images by over-parameterizing images with deepconvolutional networks. Experiments show that our method handles different testimages and varying corruption levels with a single learning pipeline where thenetwork width and termination conditions do not need to be adjusted on acase-by-case basis. Underlying the success is again the implicit bias withdiscrepant learning rates on different over-parameterized parameters, which maybear on broader applications.

Robust Recovery via Implicit Bias of Discrepant Learning Rates for Double Over-parameterization