Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery

Yangyang Xu; Junbo Ke; You-Wei Wen; Chao Wang

cs.CVMar 1, 2026

Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery

Authors: Yangyang Xu, Junbo Ke, You-Wei Wen, Chao Wang

Organizations: Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University · Department of Statistics and Data Science, Southern University of Science and Technology

Abstract

Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.

Figures & tables

Figure 1: (a) Frequency analysis of TR factors. Low-pass filtering $\{\mathcal{G}^{(k)}\}_{k=1}^{3}$ along mode-2 yields the low-frequency counterparts $\{\tilde{\mathcal{G}}^{(k)}\}_{k=1}^{3}$ . Reconstructing via the contraction $\Phi(\cdot)$ using these factors shows noticeable attenuation along the corresponding modes. (b) Qualitative comparison of TRLRF [ 45 ] , TRFD, and RepTRFD for meshgrid and non-meshgrid data recovery. Compared with TRLRF, our methods (TRFD and RepTRFD) effectively handle non-meshgrid data and demonstrate marked improvements over the baseline.

Figure 2: Illustration of proposed TRFD and RepTRFD. TRFD directly learns a continuous mapping from coordinates to TR factors. In contrast, RepTRFD reparameterizes each TR factor as a structured combination of a learnable latent tensor and a fixed basis, which improves the training dynamics and facilitates more effective learning of high-frequency components.

Panel of Figure 3 — Figure 3: Visual inpainting results on color image Airplane ( $\text{SR}=0.2$ ) and video News ( $\text{SR}=0.1$ ).

Data	Method	$\text{SR}=0.1$		$\text{SR}=0.2$		$\text{SR}=0.3$		Time (s)
Data	Method	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	Time (s)
Airplane House Pepper Sailboat $(256\times 256\times 3)$	TRLRF	17.52	0.326	21.57	0.546	24.85	0.698	5.12
	FCTN	17.38	0.359	21.23	0.566	24.67	0.710	5.44
	HLRTF	20.89	0.544	25.27	0.754	28.02	0.841	3.43
	LRTFR	21.38	0.587	25.86	0.770	28.74	0.856	6.03
	DRO-TFF	23.22	0.734	27.52	0.848	30.04	0.899	13.25

Table 1: Quantitative inpainting results on all datasets.

Panel of Figure 4 — Figure 4: Visual denoising results on MSI Face and HSI Washington DC ( $\text{SD}=0.2$ ).

Data	Method	$\text{SD}=0.1$		$\text{SD}=0.2$		$\text{SD}=0.3$		Time (s)
Data	Method	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	Time (s)
Toys Face $(256\times 256\times 31)$	LRTDTV	35.85	0.895	31.27	0.718	28.45	0.554	5.97
	DeepTensor	36.23	0.914	32.48	0.833	30.25	0.702	47.93
	HLRTF	37.09	0.917	33.28	0.828	30.88	0.726	4.55
	LRTFR	37.55	0.935	33.75	0.864	31.80	0.819	5.86
	DRO-TFF	38.67	0.959	34.74	0.911	32.91	0.881	7.61

Table 2: Average quantitative results for MSI and HSI denoising under different Gaussian noise levels.

Panel of Figure 5 — Figure 5: Visual super-resolution results at $\times 4$ scaling on the Lion and Parrot images.

Method	$\times 3$		$\times 4$		$\times 6$		Time (s)
Method	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	Time (s)
PEMLP	29.26	0.740	27.84	0.676	24.28	0.439	90.15
SIREN	31.69	0.866	29.48	0.779	27.16	0.634	125.38
Gauss	30.27	0.790	28.23	0.683	26.73	0.605	186.92
WIRE	32.43	0.869	30.21	0.788	27.88	0.683	423.27
FINER	32.09	0.858	29.84	0.778	27.61	0.666	165.08

Table 3: Super-resolution results on Lion at different scales.

Panel of Figure 6 — Figure 6: Visual comparisons on point cloud recovery for Duck ( $\text{SR}=0.15$ ) and Mario ( $\text{SR}=0.2$ ).

Method	Doll	Duck	Frog	Mario	PeterRabbit	Squirrel
PEMLP	0.112	0.064	0.058	0.091	0.081	0.084
SIREN	0.111	0.062	0.057	0.089	0.074	0.085
Gauss	0.123	0.064	0.060	0.092	0.078	0.085
WIRE	0.106	0.060	0.053	0.086	0.068	0.083
FINER	0.110	0.059	0.054	0.088	0.073	0.083
LRTFR	0.114	0.061	0.061	0.095	0.082	0.094

Table 4: NRMSE results for point cloud recovery on the SHOT dataset under

\text{SR}=0.2

Panel of Figure 7 — Figure 7: PSNR curves on the color image Airplane ( $\text{SR}=0.3$ ) (left) and MSI Toy ( $\text{SR}=0.1$ ) (right), comparing models without reparameterization (w/o Rep.), with reparameterization (w/ Rep.), and different latent sizes $R_{k}=\beta r_{k}$ for $\beta=1,3,5,10$ .

Metric	Airplane		Toy		Botswana		News
Metric	w/o Rep.	w/ Rep.	w/o Rep.	w/ Rep.	w/o Rep.	w/ Rep.	w/o Rep.	w/ Rep.
PSNR	27.41	30.45	29.41	48.67	29.21	45.27	26.48	34.90
Time (s)	12.10	13.11	13.23	14.49	40.81	42.26	14.15	15.59

Table 5: Effect of reparameterization on PSNR and runtime for different datasets with

\text{SR}=0.2

Data	Initialization Scale $a$
Data	0.01	0.05	0.165 (ours)	0.3	0.5	1
Botswana	33.86	43.74	45.27	44.81	43.15	38.37
Washington DC	33.14	46.08	47.96	47.44	45.90	38.20

Table 6: PSNR results for HSI inpainting with different basis initialization scale at

\text{SR}=0.2

Panel of Figure 8 — Figure 8: Effect of shared frequency embedding on denoising performance for the MSIs Toy (left) and Face (right) with $\text{SD}=0.2$ .

Figure 9: PSNR and FLOPs versus parameter count on Airplane (

\text{SR}=0.3

) (left) and Toy (

\text{SD}=0.2

) (right).

Panel of Figure 10 — Figure 10: Visual inpainting results on color videos Container ( $\text{SR}=0.1$ ) and Salesman ( $\text{SR}=0.2$ ).

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 17 — Figure S1 : Impact of the fixed basis strategy on denoising performance for the MSIs Toy (left) and Face (right) with $\text{SD}=0.2$ .

Panel of Figure 18 — Figure S2 : PSNR variation with TR rank and frequency scaling $\omega_{0}$ for MSI denoising on Toy (left) and Face (right) datasets with $\text{SD}=0.2$ .

Panel of Figure 19 — Figure S3 : PSNR sensitivity to hyperparameters $\beta$ and $\omega_{0}$ for inpainting on Airplane ( $\text{SR}=0.3$ ) (left) and denoising on Toy ( $\text{SD}=0.2$ ) (right).

Data	LRTFR (Baseline)	RepTRFD (w/o TV)	RepTRFD (w/ TV)
Airplane	27.21	29.37	30.45
Flowers	44.27	48.53	50.13
Botswana	41.86	44.94	45.27
Carphone	30.18	31.86	32.58

Appendix

Table S1 : PSNR comparison on image and video inpainting at

\text{SR}=0.2

under different regularization settings.

Data	LRTFR (Baseline)	RepTRFD (w/o TV)	RepTRFD (w/ TV)
Lion	28.10	30.37	31.01
Parrot	27.81	29.79	30.39

Appendix

Table S2 : PSNR comparison on

\times 4

image super-resolution under different regularization settings.

Figure S4 : Loss curve on the color image Airplane (

\text{SR}=0.3

Scheme	Airplane	Toy	Botswana	News
Kaiming	30.23	48.59	45.30	34.75
Xavier	30.45	48.67	45.27	34.90

Appendix

Table S3 : PSNR results of different variance-preserving initialization schemes on multiple datasets with

\text{SR}=0.2

Panel of Figure 24 — Figure S5 : Visual inpainting results on various datasets. From top to bottom: color images ( House , Peppers , Sailboat ) with $\text{SR}=0.2$ ; MSIs ( Toy , Flowers ) with $\text{SR}=0.05$ and $0.1$ , respectively; HSI ( Botswana ) with $\text{SR}=0.05$ ; and video sequences ( Grandma , Carphone ) with $\text{SR}=0.1$ and $0.2$ , respectively.

Data	Method	$\text{SR}=0.05$			$\text{SR}=0.1$			$\text{SR}=0.15$			$\text{SR}=0.2$			$\text{SR}=0.25$			$\text{SR}=0.3$
Data	Method	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE
Airplane	TRLRF	15.08	0.172	0.218	17.98	0.347	0.156	19.90	0.444	0.125	21.50	0.529	0.104	23.35	0.623	0.084	25.58	0.712	0.065
	FCTN	14.64	0.219	0.229	17.79	0.370	0.159	20.05	0.504	0.123	22.08	0.600	0.097	24.11	0.689	0.077	25.70	0.747	0.064
	HLRTF	18.46	0.396	0.148	21.89	0.598	0.099	24.82	0.746	0.071	26.77	0.811	0.057	27.61	0.795	0.051	29.40	0.852	0.042
	LRTFR	19.66	0.499	0.129	22.09	0.655	0.097	25.57	0.765	0.065	27.21	0.822	0.054	28.60	0.858	0.046	29.38	0.878	0.042
	DRO-TFF	21.08	0.741	0.109	22.71	0.763	0.090	26.28	0.843	0.060	27.64	0.853	0.051	29.17	0.888	0.043	30.37	0.906	0.037

Appendix

Table S4: Quantitative comparison on color images. PSNR, SSIM, and NRMSE are reported under different SRs.

Data	Method	$\text{SR}=0.03$			$\text{SR}=0.05$			$\text{SR}=0.1$			$\text{SR}=0.15$			$\text{SR}=0.2$
Data	Method	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE
Toys $(256\times 256\times 31)$	TRLRF	25.13	0.625	0.189	29.32	0.791	0.117	33.67	0.901	0.071	36.56	0.943	0.051	38.67	0.964	0.040
	FCTN	28.22	0.788	0.132	32.24	0.882	0.083	37.46	0.955	0.046	40.04	0.972	0.034	42.04	0.982	0.027
	HLRTF	30.73	0.880	0.099	33.85	0.929	0.069	40.15	0.982	0.033	42.64	0.988	0.025	45.56	0.993	0.018
	LRTFR	32.95	0.924	0.077	34.92	0.948	0.061	39.95	0.982	0.034	41.87	0.987	0.027	43.64	0.988	0.022
	DRO-TFF	33.88	0.966	0.069	37.19	0.982	0.047	40.62	0.992	0.032	43.73	0.994	0.022	45.75	0.995	0.017

Appendix

Table S5: Quantitative comparison on MSIs, HSIs and videos. PSNR, SSIM, and NRMSE are reported under different SRs.

Panel of Figure 27 — Figure S6 : Visual denoising results on MSI Toy and Balloons , and HSI KSC and Indian Pines with $\text{SD}=0.2$ .

Data	Method	$\text{SD}=0.1$			$\text{SD}=0.15$			$\text{SD}=0.2$			$\text{SD}=0.25$			$\text{SD}=0.3$
Data	Method	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE	PSNR	SSIM	NRMSE
Toys $(256\times 256\times 31)$	LRTDTV	34.85	0.906	0.062	32.17	0.824	0.084	30.29	0.734	0.104	28.79	0.653	0.124	27.65	0.581	0.141
	DeepTensor	35.11	0.902	0.060	32.50	0.875	0.081	30.91	0.807	0.097	29.50	0.760	0.114	28.62	0.672	0.126
	HLRTF	35.70	0.912	0.056	33.30	0.883	0.074	31.89	0.821	0.087	30.78	0.768	0.098	29.46	0.698	0.115
	LRTFR	36.01	0.936	0.054	34.01	0.918	0.068	32.36	0.870	0.082	31.36	0.869	0.092	30.60	0.829	0.101
	DRO-TFF	37.34	0.954	0.046	35.07	0.933	0.060	33.11	0.902	0.075	32.24	0.887	0.083	31.10	0.875	0.095

Appendix

Table S6: Quantitative comparison for denoising under different noise levels. PSNR, SSIM, and NRMSE are reported.

Panel of Figure 29 — Figure S7 : Visual super-resolution results at $\times 4$ scaling on the DIV2K dataset.

Panel of Figure 30 — Figure S8 : Visual comparisons on point cloud recovery over six objects at three SRs: $\text{SR}=0.1$ ( Frog , Duck ), $\text{SR}=0.15$ ( PetterRabbit , Mario ), and $\text{SR}=0.2$ ( Doll , Cola ).

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Tensor recovery spans heterogeneous data structures and observation processes, calling for a representation that can adapt across modalities and tasks. We propose DiffTR, an unsupervised framework that unifies tensor completion, denoising, and hyperspectral super-resolution through coordinate-conditioned diffusion. Shared factors indexed by tensor coordinates condition a compact score network, and score matching jointly learns the factors and an implicit tensor representation from the available observations. Scalar and vector-valued outputs accommodate individual entries and entire tubes, enabling a common formulation for tabular tensors, images, and videos. The framework separates representation learning from task-specific observation modeling: reverse diffusion recovers missing values; estimation via Tweedie's formula, soft thresholding, and optional spatial consistency support mixed-noise denoising; and measurement-guided sampling transfers a prior learned from low-resolution hyperspectral observations to high-resolution reconstruction, followed by post-sampling refinement with data consistency and spectral subspace regularization. This separation makes the representation compatible with different corruption models and permits task-specific constraints to be incorporated within the same recovery framework. Experiments across multiple tensor modalities and recovery tasks demonstrate competitive reconstruction quality and the flexibility of a unified conditional generative representation. The code is available at https://github.com/CZY-Code/DiffTR

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