A New Tensor Network: Tubal Tensor Train and Its Applications

TL;DR

The paper introduces the tubal tensor train (TTT) decomposition, a novel tensor-network model that efficiently combines t-product algebra with tensor train structure, enabling scalable applications in image and video processing.

Contribution

It proposes the TTT decomposition, a new tensor network model that reduces storage complexity and introduces computational strategies for practical tensor applications.

Findings

Linear storage scaling with modes for bounded tubal ranks

Effective in image and video compression tasks

Demonstrates competitive tensor completion performance

Abstract

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.