Adaptive sampling with tensor leverage scores for exact low-rank third-order tensor completion

Abstract

Tensor completion aims at estimating the missing entries from the incomplete observation. Under the tensor singular value decomposition framework, the exact recovery of a low-tubal-rank third-order tensor could be achieved via convex optimization with high probability if the tensor satisfies the tensor incoherence condition. In this work, we show that, when the random selection of entries is made adaptive to a distribution which is dependent on the coherence structure of the tensor, any low-tubal-rank tensor of the size $n\times n\times n$ with tubal-rank r can be exactly recovered with high probability from as few as $O(r{n}^2{\log }^2(n\left)\right)$ randomly chosen entries. In practice, tensor leverage scores are not known a priori, and we design a two-phase adaptive sampling strategy to obtain the leverage scores. Numerical experiments on synthetic and real-world third-order tensor data sets are used to validate our theoretical results and illustrate that the tensor recovery performance of the proposed two-phase adaptive sampling scheme is better than that of the other state-of-the-art methods.