A Fast Algorithm for Cosine Transform Based Tensor Singular Value Decomposition
This work addresses a computational bottleneck in tensor decomposition for researchers in signal processing and data analysis, offering an incremental improvement over existing methods.
The paper tackles the computational inefficiency of tensor singular value decomposition (t-SVD) by proposing a method based on the discrete cosine transform (DCT) instead of the discrete Fourier transform (DFT), resulting in a two-times faster algorithm with smaller errors in tasks like video and multispectral image completion.
Recently, there has been a lot of research into tensor singular value decomposition (t-SVD) by using discrete Fourier transform (DFT) matrix. The main aims of this paper are to propose and study tensor singular value decomposition based on the discrete cosine transform (DCT) matrix. The advantages of using DCT are that (i) the complex arithmetic is not involved in the cosine transform based tensor singular value decomposition, so the computational cost required can be saved; (ii) the intrinsic reflexive boundary condition along the tubes in the third dimension of tensors is employed, so its performance would be better than that by using the periodic boundary condition in DFT. We demonstrate that the tensor product between two tensors by using DCT can be equivalent to the multiplication between a block Toeplitz-plus-Hankel matrix and a block vector. Numerical examples of low-rank tensor completion are further given to illustrate that the efficiency by using DCT is two times faster than that by using DFT and also the errors of video and multispectral image completion by using DCT are smaller than those by using DFT.