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arXiv · 2603.10503

A New Tensor Network: Tubal Tensor Train and Its Applications

Salman Ahmadi-Asl·Valentin Leplat·Anh-Huy Phan·Andrzej Cichocki
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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.

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Salman Ahmadi-Asl, Valentin Leplat, Anh-Huy Phan, Andrzej Cichocki. 2026-03-11. A New Tensor Network: Tubal Tensor Train and Its Applications. https://arxiv.org/abs/2603.10503

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