A New Tensor Network: Tubal Tensor Train and Its Applications

2026-03-11

AI Summary

This work addresses the exponential storage growth bottleneck encountered when directly extending tensor singular value decomposition (T-SVD) to higher-order tensors, which hinders efficient processing of multi-dimensional data with tubal structures. The authors propose the Tubal Tensor Train (TTT) decomposition, which uniquely integrates t-product algebra with the low-rank structure of tensor trains. By constructing a tensor network using two third-order boundary cores and multiple fourth-order internal cores, TTT achieves linear storage complexity with respect to the number of modes. The method employs a TTT-SVD sequential fixed-rank construction algorithm and an optimization strategy based on Alternating Two-Core Updates (ATCU) in the Fourier domain. Experiments on image/video compression, tensor completion, and hyperspectral imaging demonstrate its efficacy, while theoretical analysis provides error bounds analogous to those of TT-SVD.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• tensor network
• tubal tensor
• tensor train
• t-product
• storage complexity

Innovation

• Tubal Tensor Train
• t-product
• Tensor Train decomposition
• T-SVD
• Tensor completion