# A New Tensor Network: Tubal Tensor Train and Its Applications

arXiv: https://arxiv.org/abs/2603.10503 · HTML: https://mlsanity.com/paper/2603.10503/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 32 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 5 of 5)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Hyperspectral pansharpening: a review](https://mlsanity.com/paper/1504.04531/) (2015)
- [Convolutional Tensor-Train LSTM for Spatio-temporal Learning](https://mlsanity.com/paper/2002.09131/) (2020)
- [Multi-Band Image Fusion Based on Spectral Unmixing](https://mlsanity.com/paper/1603.08720/) (2016)
- [Hot-SVD: Higher-Order t-Singular Value Decomposition for Tensors based on Tensor-Tensor Product](https://mlsanity.com/paper/2204.10229/) (2022)

## Abstract (arXiv)

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.