Tensor Subspace Detection with Tubal-Sampling and Elementwise-Sampling

The problem of testing whether an incomplete tensor lies in a given tensor subspace, called tensor matched subspace detection, is significant when it is unavoidable to have missing entries. Compared with the matrix case, the tensor matched subspace detection problem is much more challenging due to the curse of dimensionality and the intertwinement between the sampling operator and the tensor product operation. In this paper, we investigate the subspace detection problem for the transform-based tensor models. Under this framework, tensor subspaces and the orthogonal projection onto a given subspace are defined, and the energies of a tensor outside the given subspace (also called residual energy in statistics) with tubal-sampling and elementwise-sampling are derived. We have proved that the residual energy of sampling signals is bounded with high probability. Based on the residual energy, the reliable detection is feasible.

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