Tensor Decomposition Structure Search Framework from an Interaction Perspective

Ting-Wei Zhou; Xi-Le Zhao; Sheng Liu; Wei-Hao Wu; Yu-Bang Zheng; Deyu Meng

cs.CVMar 3, 2026

Tensor Decomposition Structure Search Framework from an Interaction Perspective

Authors: Ting-Wei Zhou, Xi-Le Zhao, Sheng Liu, Wei-Hao Wu, Yu-Bang Zheng, Deyu Meng

Organizations: School of Mathematical Sciences/Research Center for Image and Vision Computing, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China · School of Information Science and Technology, Southwest Jiaotong University, Chengdu 611756, China · School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China, and also with Pazhou Laboratory (Huangpu), Guangzhou, Guangdong 510335, China

Abstract

Recently, tensor decompositions have attracted increasing attention. Fundamentally, different interactions among factors induce distinct tensor decomposition structures (i.e., tensor decomposition). Identifying an appropriate interaction-induced tensor decomposition structure for given data is a fundamental yet challenging problem in tensor modeling, and remains largely under-explored. Existing tensor decomposition structure search methods are typically restricted to a predefined interaction family, such as tensor contraction. To address this problem, we suggest a tensor decomposition structure search framework (I-TSS) from an interaction perspective that can identify either a single structure beyond a predefined interaction family or a mixture of structures involving heterogeneous interaction families. Specifically, we first systematically review existing tensor decomposition structures from an interaction perspective and construct a heterogeneous interaction-induced candidate structure set. Based on it, we propose a unified energy-based rank estimation scheme for heterogeneous interaction-induced candidate structure set. We then introduce the top- $k$ gating mechanism with learnable gating scores to dynamically select a suitable single-candidate structure or combine suitable multi-candidate structures, thereby identifying the tensor decomposition structure in a data-adaptive manner. Theoretically, we derive an approximation error bound for I-TSS, thereby establishing its approximation capability. Extensive experiments on synthetic and real-world datasets demonstrate that I-TSS consistently outperforms state-of-the-art tensor decomposition methods.

Figures & tables

Characteristic	Search Capability							Data Adaptability
Method	CP	Tucker	TT	TR	FCTN	T-SVD	Mixture	Data Adaptability
Tucker	✓	✓
FCTN			✓	✓	✓
T-SVD						✓
SVDinsTN			✓	✓	✓			✓
I-TSS	✓	✓	✓	✓	✓	✓	✓	✓

TABLE I: Comparison between classic tensor decomposition structure search methods and our I-TSS in terms of search capability and data adaptability.

Fig. 1: Diagram of our I-TSS. First, we perform the energy-based rank estimation scheme for the input data. Then, the estimated ranks are fed into the heterogeneous interaction-induced candidate structure set. Finally, the top- $k$ gating mechanism is introduced to dynamically select a suitable single-candidate structure or combine suitable multi-candidate structures, leading to either a single structure ( $k=1$ ) or mixture of structures ( $k>1$ ).

Method	Rank: 3		Rank: 5		Rank: 7
Method	CR	RE	CR	RE	CR	RE
Synthetic Data (Tucker)
Tucker	0.3%	$\bf 3*10^{-5}$	0.7%	$\bf 2*10^{-5}$	1.1%	$\bf 2*10^{-5}$
TF	0.4%	$4*10^{-1}$	1.0%	$2*10^{-1}$	1.9%	$2*10^{-1}$
TNGreedy	0.4%	$6*10^{-2}$	1.5%	$9*10^{-4}$	3.0%	$8*10^{-4}$
SVDinsTN	0.6%	$8*10^{-4}$	2.2%	$5*10^{-4}$	4.3%	$2*10^{-4}$

TABLE II: Quantitative comparison of CR

(\downarrow)

and RE

(\downarrow)

of different methods on third-order synthetic data.

Method	Cloth		Beads		Jelly
Method	CR	RE	CR	RE	CR	RE
Tucker	37.8%	$1*10^{-2}$	39.3%	$1*10^{-2}$	34.5%	$1*10^{-2}$
TF	34.4%	$1*10^{-2}$	39.0%	$1*10^{-2}$	35.4%	$1*10^{-2}$
TNGreedy	31.7%	$7*10^{-3}$	37.8%	$6*10^{-3}$	32.9%	$7*10^{-3}$
SVDinsTN	32.8%	$7*10^{-3}$	39.9%	$9*10^{-3}$	34.8%	$8*10^{-3}$
I-TSS	31.7%	$\bf 6*10^{-3}$	37.8%	$\bf 5*10^{-3}$	32.6%	$\bf 6*10^{-3}$

TABLE III: Quantitative comparison of CR

(\downarrow)

and RE

(\downarrow)

of different methods on real-world MSIs.

SR		0.1			0.15			0.2			0.25			0.3
Data	Method	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE
	HaLRTC	20.54	0.672	0.592	29.89	0.830	0.202	31.73	0.864	0.163	33.50	0.888	0.133	33.86	0.895	0.127
	SiLRTCTT	28.06	0.749	0.249	31.22	0.809	0.173	33.53	0.843	0.132	35.44	0.864	0.106	36.93	0.880	0.089
Flowers	TRLRF	20.32	0.552	0.607	25.34	0.568	0.340	26.52	0.656	0.297	26.81	0.670	0.287	27.09	0.685	0.278
Flowers	HTNN	29.94	0.790	0.200	32.92	0.844	0.142	34.70	0.873	0.116	36.28	0.890	0.096	37.41	0.894	0.085
(256 $\times$ 256 $\times$ 31)	LTNN	29.74	0.784	0.205	32.95	0.840	0.142	34.81	0.869	0.114	36.32	0.886	0.096	37.52	0.898	0.083

TABLE IV: The quantitative results for real-world MSI completion by different methods. The best and second-best values are highlighted.

Panel of Figure 6 — Fig. 2: The completion results of MSIs by different methods on Flowers , Beers , Balloons , and Feathers with SR = 0.1.

SR		0.1			0.15			0.2			0.25			0.3
Data	Method	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE
	HaLRTC	18.24	0.589	0.328	22.09	0.705	0.211	24.28	0.772	0.164	25.72	0.814	0.138	27.08	0.848	0.118
	SiLRTCTT	27.18	0.851	0.117	29.84	0.903	0.086	31.81	0.931	0.068	33.36	0.947	0.057	34.67	0.959	0.049
News	TRLRF	29.75	0.861	0.087	30.11	0.863	0.083	31.02	0.882	0.075	33.10	0.919	0.059	34.82	0.942	0.048
News	HTNN	31.70	0.916	0.069	33.53	0.939	0.056	34.94	0.953	0.048	36.13	0.962	0.041	37.35	0.970	0.036
(144 $\times$ 176 $\times$ 3 $\times$ 30)	LTNN	27.24	0.860	0.116	31.29	0.929	0.073	33.54	0.952	0.056	35.13	0.964	0.047	36.58	0.972	0.039

TABLE V: The quantitative results for real-world color video completion by different methods. The best and second-best values are highlighted.

Panel of Figure 8 — Fig. 3: The completion results of real-world color videos by different methods on News , Claire , Grandma , and Akiyo with SR = 0.1.

SR		0.1			0.15			0.2			0.25			0.3
Data	Method	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE	PSNR	SSIM	RE
	HaLRTC	17.16	0.524	0.247	22.78	0.737	0.129	25.10	0.815	0.099	26.57	0.856	0.083	27.94	0.888	0.071
	SiLRTCTT	27.53	0.880	0.074	28.18	0.887	0.069	30.07	0.921	0.055	31.67	0.943	0.046	33.15	0.957	0.039
Greek	TRLRF	24.37	0.715	0.107	28.70	0.869	0.065	29.05	0.876	0.062	29.41	0.887	0.060	29.76	0.890	0.057
Greek	HTNN	29.06	0.890	0.062	31.07	0.927	0.049	32.95	0.950	0.040	34.67	0.965	0.032	36.39	0.975	0.027
(128 $\times$ 128 $\times$ 3 $\times$ 30)	LTNN	29.37	0.909	0.060	32.28	0.952	0.043	34.86	0.972	0.032	36.85	0.982	0.025	38.62	0.987	0.020

TABLE VI: The quantitative results for real-world light field data completion by different methods. The best and second-best values are highlighted.

Panel of Figure 10 — Fig. 4: The completion results of light field data by different methods on Greek , Museum , Medieval2 , and Vinyl with SR = 0.1.

Panel of Table 11 — TABLE VII: The PSNR values and corresponding interaction families of the representative search method (i.e., SVDinsTN) and the proposed I-TSS across various data with a sampling rate of 0.2. The symbol ‘+’ represents the mixture of structures.

Data	Flowers		Balloons		Feathers
Selection Strategy	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
Max Sampling	42.51	0.985	49.49	0.996	41.54	0.980
Random Sampling	37.64	0.880	40.70	0.973	37.97	0.949

TABLE VIII: The proposed I-TSS variants with different candidate structure selection strategies for MSIs Flowers , Balloons , and Feathers with a sampling rate of 0.2.

Data	Flowers		Balloons		Feathers
Updating Order	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
I	42.99	0.938	46.76	0.994	41.65	0.981
II	42.70	0.939	49.49	0.996	42.54	0.984
III	34.34	0.895	41.63	0.983	35.94	0.941

TABLE IX: The proposed I-TSS variants with different parameters updating orders for MSIs Flowers , Balloons , and Feathers with a sampling rate of 0.2.

Panel of Figure 14 — Fig. 5: The PSNR values of the proposed I-TSS with different values of energy ratio thresholds for MSIs Flowers and Feathers with a sampling rate of 0.1.

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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-

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