Orthogonal tubal rank-1 tensor pursuit for tensor completion

Abstract This work addresses the issue of tensor completion. The properties of the tensor tubal rank are firstly discussed. It is shown that the tensor tubal rank has similar properties like that of matrix rank derived from SVD. The completion algorithm for the case that the measurements are noise-free or corrupted by Gaussian noise is then proposed based on an orthogonal pursuit on tubal rank-1 tensors. The philosophy behind the devised approach is to relax the problem of tensor tubal rank minimization into tensor Frobenius-norm optimization with a constraint on the maximum number of orthogonal tensors. An iterative procedure which calculates one orthogonal tensor at each iterative step is then suggested, and the local convergence under the noise-free case is also proved. Furthermore, the proposed method is generalized to the situation where the observations are corrupted by impulsive noise in a tubal form. To tackle the impulsive noise, we formulate the problem of tensor completion as minimization of tensor tubal lp-norm with 1

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