Tensor Matched Subspace Detection

The problem of testing whether an incomplete tensor lies in a given tensor subspace is significant when it is impossible to observe all entries of a tensor. We focus on this problem under tubal-sampling and elementwise-sampling. Different from the matrix case, our problem is much more challenging due to the curse of dimensionality and different definitions for tensor operators. In this paper, the problem of matched subspace detections are discussed based on tensor product (t-product) and tensor-tensor product with invertible linear transformation (L-product). Based on t-product and L-product, a tensor subspace can be defined, and the energy of a tensor outside the given subspace (also called residual energy in statistics) is bounded with high probability based on samples. For a tensor in R13 , the reliable detection is possible when the number of its elements we obtained is slightly greater than r × n3 both with t-product and L-product, where r is the dimension of the given tensor subspace.