Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization

Biswarup Karmakar; Ratikanta Behera

stat.MLApr 15, 2026

Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization

Authors: Biswarup Karmakar, Ratikanta Behera

Organizations: Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, 560012, India.

Abstract

The robust low-rank tensor completion problem addresses the challenge of recovering corrupted high-dimensional tensor data with missing entries, outliers, and sparse noise commonly found in real-world applications. Existing methodologies have encountered fundamental limitations due to their reliance on uniform regularization schemes, particularly the tensor nuclear norm and $\ell_1$ norm regularization approaches, which indiscriminately apply equal shrinkage to all singular values and sparse components, thereby compromising the preservation of critical tensor structures. The proposed tensor weighted correlated total variation (TWCTV) regularizer addresses these shortcomings through an $M$ -product framework that combines a weighted Schatten- $p$ norm on gradient tensors for low-rankness with smoothness enforcement and weighted sparse components for noise suppression. The proposed weighting scheme adaptively reduces the thresholding level to preserve both dominant singular values and sparse components, thus improving the reconstruction of critical structural elements and nuanced details in the recovered signal. Through a systematic algorithmic approach, we introduce an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical substantiation, with convergence properties comprehensively analyzed within the $M$ -product framework.Comprehensive numerical evaluations across image completion, denoising, and background subtraction tasks validate the superior performance of this approach relative to established benchmark methods.

Figures & tables

Panel of Figure 1 — Figure 1: Flowchart of the model for tensor completion and TRPCA.

$n$	$r$	DCT		DFT		DWT		ROT
$n$	$r$	RE	Time	RE	Time	RE	Time	RE	Time
30	3	$1.78\times 10^{-8}$	7.56	$7.43\times 10^{-8}$	25.69	$8.41\times 10^{-6}$	29.51	$8.45\times 10^{-6}$	29.39
30	5	$3.97\times 10^{-8}$	7.53	$7.84\times 10^{-8}$	25.75	$8.46\times 10^{-6}$	29.52	$8.47\times 10^{-6}$	29.23
50	3	$1.09\times 10^{-7}$	18.71	$6.83\times 10^{-7}$	27.36	$8.37\times 10^{-6}$	31.78	$8.40\times 10^{-6}$	31.96
	5	$1.74\times 10^{-7}$	19.01	$7.43\times 10^{-7}$	27.39	$8.42\times 10^{-6}$	32.16	$8.44\times 10^{-6}$	31.86
	8	$3.79\times 10^{-7}$	19.12	$7.81\times 10^{-7}$	28.80	$3.22\times 10^{-5}$	35.01	$8.48\times 10^{-6}$	34.97

Table 1: Recovered results on fourth-order synthetic tensor (

d=4

n_{3}=n_{4}=20

p=0.5

Panel of Figure 2 — Figure 2: Relative error curves showing convergence behavior of: (a) Tensor completion (on the tensor of size $256\times 256\times 3$ ) with TWCTV regularization and (b) TRPCA (on the tensor of size $256\times 256\times 3$ ) with TWCTV regularization and weighted $\ell_{1}$ sparse term.

Panel of Figure 3 — Figure 3: Phase transition diagrams for tensor completion (size: $40\times 40\times 20$ ): (a) TCTV regularizer with successful cases (white): 142 (37.37%) and (b) Weighted Schatten $p$ -norm ( $p=0.9$ ) based TWCTV regularizer with successful cases (white): 160 (42.11%).

Panel of Figure 4 — Figure 4: Sensitivity analysis of the Schatten- $p$ exponent on synthetic tensor data ( $80\times 80\times 20$ , sampling rate $8\%$ , corruption ratio $2\%$ ).

Tensor Size	Model	Solver Residual	Time (s)	Iterations
$80\times 80\times 20$	Two Nonconvex	$2.82\times 10^{-9}$	$5.06$	$170$
$80\times 80\times 20$	One Nonconvex	$1.36\times 10^{-8}$	$4.99$	$164$
$100\times 100\times 20$	Two Nonconvex	$6.65\times 10^{-9}$	$8.22$	$175$
$100\times 100\times 20$	One Nonconvex	$1.92\times 10^{-8}$	$8.18$	$173$

Table 2: Ablation study comparing two-nonconvex and one-nonconvex models on synthetic tensors. Solver residuals measured as

\|P_{\Omega}(\mathcal{M})-\mathcal{X}-\mathcal{E}\|_{F}

Panel of Figure 5 — Figure 5: Visual comparison of different tensor completion methods with an observed tensor having 95% missing entries.

Image	SR = 5%				SR = 10%
Image	PSNR	SSIM	FSIM	Time	PSNR	SSIM	FSIM	Time
Observed	5.966	0.013	0.490	0.000	6.199	0.025	0.559	0.000
SNN	18.809	0.344	0.547	2.860	21.594	0.548	0.701	2.888
SNN+TV	22.265	0.684	0.833	11.686	24.871	0.793	0.884	12.432
BCPF	23.295	0.707	0.850	19.196	25.570	0.796	0.903	33.455
KBR	20.317	0.580	0.797	19.188	25.489	0.797	0.900	22.017

Table 3: Results for color image inpainting under different sampling rates (SR).

Panel of Figure 6 — Figure 6: Visual comparison of different tensor completion methods under a text mask. The first row shows the observed image and results from SNN, BCPF, KBR, and TNN. The second row shows results from IRTNN, SPC+TV, TNN+TV, TCTV, and the proposed TWCTV method.

Image	PSNR	SSIM	FSIM	Time
Observed	14.957	0.786	0.772	0.000
SNN	33.706	0.977	0.983	1.801
BCPF	29.883	0.948	0.967	55.891
KBR	34.547	0.978	0.986	20.953
TNN	28.966	0.928	0.945	4.180
IRTNN	33.242	0.973	0.982	55.065

Table 4: Quantitative comparison of different methods on color image inpainting using text mask.

Panel of Figure 7 — Figure 7: Pseudo-color visualization of MSI completion methods using bands (R:25, G:15, B:5).

Image	PSNR	SSIM	ERGAS
Observed	17.956	0.230	544.529
SNN	20.665	0.331	395.775
KBR	25.133	0.624	232.986
TNN	21.608	0.388	361.431
TNN+TV	22.249	0.423	321.439
TCTV	25.096	0.661	231.123

Table 5: Quantitative results for MSI image inpainting with 5% sampling rate.

Panel of Figure 8 — Figure 8: Parameter sensitivity for the penalty parameter $\rho$ for TWCTV based tensor completion on MSI image (size: $256\times 256\times 31$ ) with 80% missing entries.

Panel of Figure 9 — Figure 9: Visual comparison of different tensor RPCA methods on “starfish”. The first row shows the original image, observed noisy tensor (Noise level = 0.5), and results from SNN, KBR, and LRTV. The second row shows results from LRTDTV, TLR-HTV, TNN, TCTV, and the proposed TWCTV method.

Method	Noise = 0.5			Noise = 0.3			Noise = 0.1
Method	PSNR	SSIM	FSIM	PSNR	SSIM	FSIM	PSNR	SSIM	FSIM
Noisy	7.803	0.084	0.454	10.012	0.155	0.535	14.803	0.369	0.738
SNN	18.341	0.397	0.742	24.552	0.796	0.887	27.996	0.917	0.944
KBR	17.890	0.324	0.700	21.271	0.678	0.845	23.929	0.883	0.931
LRTV	19.328	0.506	0.774	24.085	0.748	0.884	29.381	0.916	0.955
LRTDTV	18.012	0.459	0.745	22.602	0.715	0.868	29.209	0.899	0.947

Table 6: Comparison of different methods on color image denoising under different levels of salt-and-pepper noise.

Panel of Figure 10 — Figure 10: Parameter sensitivity for the parameter $\rho$ for TWCTV based TRPCA on color image (size: $256\times 256\times 3$ ) with 50% impulse noise or salt-paper noise.

Panel of Figure 11 — Figure 11: Visual comparison of different methods for HSI denoising with a sparse noise rate of 30%.

Method	SER = 10%			SER = 30%			SER = 50%
Method	PSNR	ERGAS	SSIM	PSNR	ERGAS	SSIM	PSNR	ERGAS	SSIM
Noisy	14.524	699.002	0.275	9.752	1210.860	0.075	7.526	1564.154	0.032
SNN	43.328	35.941	0.997	38.694	58.770	0.990	33.838	94.077	0.971
KBR	34.127	77.090	0.970	31.593	100.917	0.949	24.538	216.427	0.709
LRTV	38.536	44.482	0.985	34.863	68.429	0.964	30.786	109.954	0.909
LRTDTV	38.181	46.094	0.979	36.865	53.601	0.974	33.734	77.380	0.950

Table 7: Performance comparison of various methods for HSI denoising under different sparse error rates (SER).

Panel of Figure 12 — Figure 12: Foreground detection results on frames 900 (top) and 925 (bottom) comparing different methods.

Dataset	Metrics	RPCA	SNN	TNN	TCTV	TWCTV
highway	Precision	0.8873	0.8966	0.8566	0.8232	0.8990
	Recall	0.7886	0.7092	0.8690	0.7992	0.8866
	F-Measure	0.8348	0.7923	0.8522	0.8110	0.8927

Table 8: Performance comparison of foreground detection on the CDNet dataset.

Tensor Completion

Tensor completion reconstructs missing entries in multiway data by exploiting low-rank structure and dependencies across modes.

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Robust low-rank tensor completion via factorized weighted tensor schatten-p norm minimization

Binghao Wang, Feng Zhang, Wendong Wang, Jianjun Wang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China · College of Artificial Intelligence, Southwest University, Chongqing 400715, China

Tensor Completion Tensor Decomposition Nonconvex Optimization

Open paper AskPDF arXiv

Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-

p

tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-

p

norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)--block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush--Kuhn--Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-

δ

smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.

Abstract

Tensor Completion

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