Coherent Low-tubal-Rank Tensor Completion

Abstract

The sufficient condition of exact completion of coherent low-tubal-rank tensors is studied in this paper. When the leveraged sampling strategy is adopted instead of the uniform sampling strategy, it can be shown that any 3-D tensor of size n_1 × n_2 × n_3 having tubal-rank r can be exactly recovered using tubal nuclear norm minimization with high probability when the number of observed entries is of order O(max{n_1, n_2}n_3r log^2((n_1+n_2)n_3)). This result removes the tensor incoherence parameter μ_0 in the sample complexity O(μ_0 max{n_1, n_2}n_3r log((n_1+n_2)n_3)) of uniform sampling strategy and can significantly reduce the number of observations for a tensor with The sufficient condition of exact completion of coherent low-tubal-rank tensors is studied in this paper. When the leveraged sampling strategy is adopted instead of the uniform sampling strategy, it can be shown that any 3-D tensor of size n1 x n2 x n3 having tubal-rank r can be exactly recovered using tubal nuclear norm minimization with high probability when the number of observed entries is of order O(max{n1, n2}n3r log2((n1+n2)n3)). This result removes the tensor incoherence parameter µ0 in the sample complexity O(µ0 max{n1, n2}n3r log((n1+n2)n3)) of uniform sampling strategy and can significantly reduce the number of observations for a tensor with large µ0.