Noisy tensor recovery via nonconvex optimization with theoretical recoverability

Noisy tensor recovery aims to estimate underlying low-rank tensors from the noisy observations. Besides the sparse noise, tensor data can also be corrupted by the small dense noise. Existing methods typically use the Frobenius norm to handle the small dense noise. In this work, we build a new nonconvex model to decompose the low-rank and sparse components. To be specific, we employ the ${\ell }_1$ norm to handle the small dense noise term, the ${\ell }_0$ ‘norm’ to enforce the sparse outliers, and the tensor nuclear norm to model the underlying low-rank tensor. We develop an effective alternating minimization-based algorithm. Under certain conditions, we prove that the proposed method has a high probability of exactly recovering low-rank and sparse tensors. Numerical experiments showcase the advantage of our method.