This paper considers the problem of recovering multidimensional array, in particular third-order tensor, from a random subset of its arbitrarily corrupted entries. Our study is based on a recently proposed algebraic framework in which the tensor-SVD is introduced to capture the low-tubal-rank structure in tensor. We analyze the performance of a convex program, which minimizes a weighted combination of the tensor nuclear norm, a convex surrogate for the tensor tubal rank, and the tensor l1 norm. We prove that under certain incoherence conditions, this program can recover the tensor exactly with overwhelming probability, provided that its tubal rank is not too large and that the corruptions are reasonably sparse. The number of required observations is order optimal (up to a logarithm factor) when comparing with the degrees of freedom of the low-tubal-rank tensor. Numerical experiments verify our theoretical results and real-world applications demonstrate the effectiveness of our algorithm.
Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion
