Tensor CUR Decomposition under T-Product and Its Perturbation

In order to process the large-scale data, a useful tool in dimensionality reduction of a matrix, the CUR decomposition has been developed, which can compress the huge matrix with its original elements. Tensor-tensor decompositions have become prevalent and a new multiplication of a tensor based on the T-product has been presented for the tensor computation. Using the T-product, we propose a dimensionality reduction tool of three-order tensor called the T-product CUR decomposition (t-CUR decomposition for short) and analyze its stability of the perturbation. The t-CUR decomposition can reduce the size of a large-scale tensor with its original entries, its perturbation error bound is refined in the first order of the noise tensor under the spectrum norm. Numerical tests are provided to verify the results of our theoretical error analysis as well.