Recovering low‐rank tensor from limited coefficients in any ortho‐normal basis using tensor‐singular value decomposition

Abstract Tensor singular value decomposition (t‐SVD) provides a novel way to decompose a tensor. It has been employed mostly in recovering missing tensor entries from the observed tensor entries. The problem of applying t‐SVD to recover tensors from limited coefficients in any given ortho‐normal basis is addressed. We prove that an n × n × n3 tensor with tubal‐rank r can be efficiently reconstructed by minimising its tubal nuclear norm from its O(rn3n log2(n3n)) randomly sampled coefficients w.r.t any given ortho‐normal basis. In our proof, we extend the matrix coherent conditions to tensor coherent conditions. We first prove the theorem belonging to the case of Fourier‐type basis under certain coherent conditions. Then, we prove that our results hold for any ortho‐normal basis meeting the conditions. Our work covers the existing t‐SVD‐based tensor completion problem as a special case. We conduct numerical experiments on random tensors and dynamic magnetic resonance images (d‐MRI) to demonstrate the performance of the proposed methods