A New Tensor Network: Tubal Tensor Train and Its Applications

Salman Ahmadi-Asl; Valentin Leplat; Anh-Huy Phan; Andrzej Cichocki

math.NAMar 11, 2026

A New Tensor Network: Tubal Tensor Train and Its Applications

Authors: Salman Ahmadi-Asl, Valentin Leplat, Anh-Huy Phan, Andrzej Cichocki

Organizations: Lab of Machine Learning and Knowledge Representation, Innopolis University, Innopolis, Russia · Innopolis University, Innopolis, Russia · Skolkovo Institute of Science and Technology, Center for Artificial Intelligence Technology, Moscow, Russia · Systems Research Institute of Polish Academy of Science, Warsaw, Poland

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.

Figures & tables

Figure 1: T-SVD and truncated T-SVD for a third-order tensor.

Figure 2: Tubal tensor train structure and tube-wise evaluation.

\widehat{\underline{\mathbf{X}}}(:,:,\dots,:,k)\approx\ll\mathbf{C}_{k}^{(1)},\mathbf{C}_{k}^{(2)},\dots,\mathbf{C}_{k}^{(N)}\gg

Algorithm 4 TATCU (Fourier-slice alternating two-cores update)

Figure 3: Benchmark images used in our experiments.

	TT-Based			Proposed method
Images	MSE	PSNR (dB)	SSIM	MSE	PSNR (dB)	SSIM
Kodim03	251	24.12	0.6649	114.09	27.52	0.7806
Kodim23	251.40	24.13	0.7042	127.27	27.08	0.8096
Kodim04	239.25	24.34	0.5940	93.46	28.42	0.7117
Airplane	874.30	18.71	0.6081	351.47	22.67	0.6921
Kodim15	380.32	22.33	0.6178	159.13	26.11	0.7673

Table 1: The experimental results obtained by the proposed algorithm and the TT-based method for images and error bound of

0.15

Figure 4: The reconstructed images using the proposed TTT model and the TT-based model for an upper error bound of $0.15$ .

Method	Airplane	Barbara	Kodim02	Kodim03	Kodim04	Kodim15	Kodim20
TTT	22.67	27.30	29.90	27.52	28.42	26.11	23.35
TT	18.71	23.37	26.62	24.12	24.34	22.33	21.73
TC	20.85	26.01	27.78	26.34	26.75	25.04	21.17

Table 2: PSNR values obtained by TTT, TT, and TC for tolerance

0.15

on seven representative benchmark RGB images.

	TT-based model			Proposed method
Video	Rel. err.	Time (s)	Comp. factor	Rel. err.	Time (s)	Comp. factor
News_qcif	0.1	23.80	13.34	0.1	21.73	12.44
Akiyo_qcif	0.1	27.49	7.68	0.1	22.30	64.64
Tempete_cif	0.1	32.16	2.3283	0.1	30.33	2.3284

Table 3: Experimental results for the proposed TTT algorithm and the TT-based baseline on the video data sets.

	T-SVD			Proposed method
Video	Rel. err.	Time (s)	Comp. factor	Rel. err.	Time (s)	Comp. factor
Waterfall_cif	0.1	10.34	5.04	0.1	20.73	20.10
Foreman_qcif	0.1	9.34	4.42	0.1	24.56	11.44
Stephan_qcif	0.1	11.28	2.50	0.1	24.90	2.53

Table 4: Experimental results for the proposed TTT algorithm and the T-SVD baseline on the video data sets.

Panel of Figure 5 — Figure 5: The PSNR and SSIM of the reconstructed frames by the proposed TTT model and TT decomposition for the relative approximation error bound of $0.1$ for the “Akiyo” video (left) and the “News” video datasets.

Figure 6: Illustration of the reference data, the incomplete observations with $70\%$ of entries removed at random, and the reconstructions obtained with the proposed model and T-SVD for Example 3 . The displayed triplets correspond to ( MSE, PSNR, SSIM ).

Method	Runtime (sec.)	PSNR (dB)	RMSE	ERGAS	SAM	UIQI	#Paras
Best value	0	$\infty$	0	0	0	1	0
Data set - ROSIS Pavia Univ. - $N=14$ - Relative error fixed and set to 0.08
TTT	61.26	35.89	130.02	10.77	3.01	0.97	12454744
TT	42.93	35.89	130.03	10.71	2.82	0.97	13600746
Data set - ROSIS Pavia Univ. - $N=14$ - "equal" number of parameters
TTT	52.78	36.33	123.52	10.26	2.84	0.97	13565592

Table 5: Comparison of tensor decomposition models on the ROSIS Pavia Univ. data set. The table reports the quantitative quality detailed in Section A.2.1 .

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Runtime (sec.)	PSNR (dB)	RMSE	ERGAS	SAM	UIQI	#Paras
Best value	0	$\infty$	0	0	0	1	0
Data set - HYDICE Urban - $N=13$ - Relative error fixed and set to 0.08
TTT	39.12	33.67	16.41	10.31	3.18	0.97	9010360
TT	22.04	33.80	16.42	10.08	3.02	0.98	10670310
Data set - HYDICE Urban - $N=13$ - "equal" number of parameters
TTT	23.81	35.92	12.71	7.95	2.60	0.98	10559780

Appendix

Table 6: Comparison of tensor decomposition models on the HYDICE Urban data set. The table reports the quantitative quality detailed in Section A.2.1 .

Method	Runtime (sec.)	PSNR (dB)	RMSE	ERGAS	SAM	UIQI	#Paras
Best value	0	$\infty$	0	0	0	1	0
Data set - Moffett - $N=10$ - Relative error fixed and set to 0.02
TTT	0.68	39.72	0.01	4.90	6.22	1.00	362150
TT	0.28	40.61	0.01	3.10	5.78	1.00	418905
Data set - Moffett - $N=10$ - "equal" number of parameters
TTT	0.66	44.15	0.00	2.93	4.06	1.00	407975

Appendix

Table 7: Comparison of tensor decomposition models on the AVIRIS Moffett data set. The table reports the quantitative quality detailed in Section A.2.1 .

Method	Runtime (sec.)	PSNR (dB)	RMSE	ERGAS	SAM	UIQI	#Paras
Best value	0	$\infty$	0	0	0	1	0
Data set - K.S.C. - $N=14$ - Relative error fixed and set to 0.05
TTT	19.39	39.99	130.13	86.88	25.16	0.49	8534545
TT	26.90	38.28	130.35	119.49	37.73	0.32	3484834
Data set - K.S.C. - $N=14$ - Relative error fixed and set to 0.02
TTT	19.28	45.83	51.52	47.42	16.73	0.58	9188870

Appendix

Table 8: Comparison of tensor decomposition models on the AVIRIS Kennedy Space Center data set. The table reports the quantitative quality detailed in Section A.2.1 .

Panel of Figure 7 — Figure 7: SAM maps for HYDICE Urban data set for both stages of the test procedure detailed in Section A.2 , the values for sam quality measurement are limited to the interval [0;20].

Panel of Figure 8 — Figure 8: SAM maps for ROSIS Pavia University data set for both stages of the test procedure detailed in Section A.2 , the values for sam quality measurement are limited to the interval [0;20].

Panel of Figure 9 — Figure 9: SAM maps for AVIRIS Moffett data set for both stages of the test procedure detailed in Section A.2 , the values for sam quality measurement are limited to the interval [0;20].

Figure 10: Structure of the TT decomposition for an $N$ -th order tensor.

Figure 11: The truncated T-SVD of a tensor $\underline{\bf X}$ as a sum of tubal rank-1 terms.

Figure 12: The truncated T-SVD of a third-order and a fourth-order tensor.

Figure 13: The procedure of the TTT-SVD for decomposing a tensor into the TTT format.

Figure 14: (Upper) The visualization of the structure of the proposed tubal TC/TR decomposition. The connection between the core tensors is the T-product. (Bottom) A tube of the original tensor is obtained through a sequence of t-products involving sub-tensors of the TTC/TTR cores.

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Tensor singular value decomposition (T-SVD), which is built upon the tensor-tensor product (t-product), has emerged as a powerful tool for processing high-dimensional visual data such as color images and videos. However, the standard t-product imposes strict dimensional compatibility constraints. Although extensions based on the semi-tensor product (STP) relax this restriction, their single-term formulations still suffer from limited approximation accuracy. Moreover, these deterministic methods incur high computational costs when processing large-scale tensor data. To address these issues, this paper introduces a novel semi-tensor product for third-order tensors under the t-product framework induced by arbitrary invertible linear transforms. The resulting tensor semi-tensor product breaks the rigid dimension matching requirement of the standard t-product, while retaining the closed-form property of T-SVD. Based on this construction, we develop a multi-term semi-tensor product singular value decomposition (MSTP-SVD), which integrates multiple orthogonal decomposition terms to significantly improve low-rank approximation accuracy compared with single-term schemes. To reduce the computational cost of multi-term modeling, we incorporate randomized projection and power iteration techniques into the MSTP-SVD framework, yielding an accelerated multi-term randomized semi-tensor product SVD (MRSTP-SVD) algorithm that achieves a balance between reconstruction accuracy and computational efficiency. Experiments on image and video compression and completion tasks demonstrate the effectiveness of the proposed method.

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