A New Tensor Network: Tubal Tensor Train and Its Applications
This work addresses efficient tensor representation for applications such as data compression and imaging, but it appears incremental as it builds on existing tensor decomposition methods.
The authors introduced the tubal tensor train (TTT) decomposition, a tensor-network model combining t-product algebra with tensor train structure to achieve linear storage scaling with modes for bounded tubal ranks, and demonstrated its performance on tasks like image compression and tensor completion.
We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-$(N+1)$ tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and $N-2$ fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.