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

Binghao Wang; Feng Zhang; Wendong Wang; Jianjun Wang

cs.CVSep 13, 2026

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

Authors: Binghao Wang, Feng Zhang, Wendong Wang, Jianjun Wang

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

Abstract

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.

Figures & tables

Method	Norm / regularizer factorization	Flexible Schatten- $p$	Weighted	SVD-free	Rank adaptive
Cabral et al. [ 15 ]	$\begin{aligned} \\|\mathbf{X}\\|_{*}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}^{\top}}\frac{1}{2}\bigl(\\|\mathbf{U}\\|_{F}^{2}+\\|\mathbf{V}\\|_{F}^{2}\bigr)\end{aligned}$
RBF [ 40 ]	$\begin{aligned} \\|\mathbf{L}\\|_{}&=\\|\mathbf{U}\mathbf{V}^{\top}\\|_{}=\\|\mathbf{V}\\|_{*},\mathbf{U}^{\top}\mathbf{U}=\mathbf{I}\end{aligned}$				$\checkmark$
Bi-trace / Tri-trace [ 3 ]	$\begin{aligned} \\|\mathbf{X}\\|_{S_{1/2}}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}^{\top}}\\|\mathbf{U}\\|_{}\\|\mathbf{V}\\|_{},\\ \\|\mathbf{X}\\|_{S_{1/3}}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}\mathbf{W}^{\top}}\\|\mathbf{U}\\|_{}\\|\mathbf{V}\\|_{}\\|\mathbf{W}\\|_{*}\end{aligned}$
F/N hybrid [ 9 ]	$\begin{aligned} \\|\mathbf{X}\\|_{S_{2/3}}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}^{\top}}\\|\mathbf{U}\\|_{F}\\|\mathbf{V}\\|_{*}\end{aligned}$
Bi-Schatten- $p$ norm surrogate [ 4 ]	$\begin{aligned} \frac{1}{p}\\|\mathbf{X}\\|_{S_{p}}^{p}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}^{\top}}\left(\frac{1}{p_{1}}\\|\mathbf{U}\\|_{S_{p_{1}}}^{p_{1}}+\frac{1}{p_{2}}\\|\mathbf{V}\\|_{S_{p_{2}}}^{p_{2}}\right),\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\end{aligned}$	$\checkmark$
Unified scalable Schatten formulation [ 17 ]	$\begin{aligned} \\|\mathbf{X}\\|_{S_{p}}&=\min_{\mathbf{X}=\mathbf{U}\mathbf{V}^{\top}}\\|\mathbf{U}\\|_{S_{p_{1}}}\\|\mathbf{V}\\|_{S_{p_{2}}},\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\end{aligned}$	$\checkmark$

TABLE I: Comparison of representative factorized low-rank norm and variational models discussed in Section II-B.

Compared Methods	Algorithm Formulation
RPCA [ 29 ]	$\displaystyle\begin{aligned} \min_{\mathbf{X},\mathbf{N}}\;&\lambda\\|\mathbf{X}\\|_{*}+\\|\mathbf{M}\odot\mathbf{N}\\|_{1},\\[-2.84526pt] &\mathrm{s.t.}\;\mathbf{Y}=\mathbf{X}+\mathbf{N}\end{aligned}$
UNIFY [ 15 ]	$\displaystyle\begin{aligned} \min_{\mathbf{U},\mathbf{V},\mathbf{X}}\;&\frac{\lambda}{2}\left(\\|\mathbf{U}\\|_{F}^{2}+\\|\mathbf{V}\\|_{F}^{2}\right)+\\|\mathbf{M}\odot(\mathbf{Y}-\mathbf{X})\\|_{1},\\[-2.84526pt] &\mathrm{s.t.}\;\mathbf{X}=\mathbf{U}\mathbf{V}\end{aligned}$
GIRNN [ 34 ]	$\displaystyle\begin{aligned} \min_{\mathbf{X},\mathbf{N}}\;&\lambda\\|\mathbf{X}\\|_{w,*}+\\|\mathbf{M}\odot\mathbf{N}\\|_{1},\\[-2.84526pt] &\mathrm{s.t.}\;\mathbf{Y}=\mathbf{X}+\mathbf{N}\end{aligned}$
RLMF [ 19 ]	$\displaystyle\begin{aligned} \min_{\mathbf{U},\mathbf{V}}\;&\frac{\lambda}{2}\left(\\|\mathbf{U}\\|_{w,F}^{2}+\\|\mathbf{V}\\|_{w,F}^{2}\right)\\[-2.84526pt] &+\\|\mathbf{M}\odot(\mathbf{U}\mathbf{V}-\mathbf{Y})\\|_{1}\end{aligned}$

TABLE II: Matrix-based baseline methods and their objective functions.

Compared Methods	Algorithm Formulation
SNN [ 49 ]	$\displaystyle\begin{aligned} \min_{\mathcal{X},\mathcal{N}}\;&\sum_{i=1}^{3}\lambda_{i}\\|\mathbf{X}_{(i)}\\|_{*}+\\|\mathcal{M}\odot\mathcal{N}\\|_{1},\\[-2.84526pt] &\mathrm{s.t.}\;\mathcal{Y}=\mathcal{X}+\mathcal{N}\end{aligned}$
TNN [ 8 ]	$\displaystyle\begin{aligned} \min_{\mathcal{X},\mathcal{N}}\;&\lambda\\|\mathcal{X}\\|_{*}+\\|\mathcal{M}\odot\mathcal{N}\\|_{1},\\[-2.84526pt] &\mathrm{s.t.}\;\mathcal{Y}=\mathcal{X}+\mathcal{N}\end{aligned}$
RLTF [ 19 ]	$\displaystyle\begin{aligned} \min_{\mathcal{U},\mathcal{V}}\;&\frac{\lambda}{2}\left(\\|\mathcal{U}\\|_{w,F}^{2}+\\|\mathcal{V}\\|_{w,F}^{2}\right)\\[-2.84526pt] &+\\|\mathcal{M}\odot(\mathcal{U}*\mathcal{V}-\mathcal{Y})\\|_{1}\end{aligned}$
WNNTF	$\displaystyle\begin{aligned} \min_{\mathcal{U},\mathcal{V},\mathcal{X}}\;&\\|\mathcal{P}_{\Omega}(\mathcal{Y}-\mathcal{X})\\|_{1}+\frac{\lambda}{2}\left(\\|\mathcal{U}\\|_{W,}+\\|\mathcal{V}\\|_{W,}\right),\\[-2.84526pt] &\mathrm{s.t.}\;\mathcal{X}=\mathcal{U}*\mathcal{V}\end{aligned}$
SPTF	$\displaystyle\begin{aligned} \min_{\mathcal{U},\mathcal{V},\mathcal{X}}\;&\\|\mathcal{P}_{\Omega}(\mathcal{Y}-\mathcal{X})\\|_{1}+\frac{\lambda}{2}\left(\\|\mathcal{U}\\|_{S_{p_{1}}}^{p_{1}}+\\|\mathcal{V}\\|_{S_{p_{2}}}^{p_{2}}\right),\\[-2.84526pt] &\mathrm{s.t.}\;\mathcal{X}=\mathcal{U}*\mathcal{V},\hskip 8.19447pt\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\end{aligned}$
WTSP [ 44 ]	$\displaystyle\begin{aligned} \min_{\mathcal{X}}\;&\\|\mathcal{P}_{\Omega}(\mathcal{Y}-\mathcal{X})\\|_{1}\\[-2.84526pt] &+\lambda\\|\mathcal{X}\\|_{W,S_{p}}^{p}\end{aligned}$

TABLE III: Tensor-based baseline methods and their objective functions.

Method Type	Method	Gaussian Noise			GMM Noise			Laplace Noise
Method Type	Method	PSNR	SSIM	Time (s)	PSNR	SSIM	Time (s)	PSNR	SSIM	Time (s)
Matrix-Based Methods	RPCA	22.11	0.78	1.65	27.44	0.92	1.62	23.64	0.83	1.70
	Unifying	22.47	0.79	0.29	28.47	0.94	0.31	23.99	0.84	0.32
	GIRNN	22.42	0.79	1.71	27.87	0.93	1.68	23.84	0.84	1.75
	RLMF	24.98	0.87	0.25	27.05	0.91	0.33	25.24	0.88	0.31
	WS p MF-I (Ours)	25.08	0.87	0.29	28.70	0.94	0.37	26.04	0.89	0.37

TABLE IV: Quantitative results for synthetic tensor completion under different noise distributions. For the reconstruction-quality metrics, the best and second-best results among all methods are marked in bold and underlined , respectively. Lower running times are better.

Fig. 2: PSNR trajectories of the matrix- and tensor-based methods over

150

outer iterations under Gaussian, GMM, and Laplace noise. The horizontal and vertical axes denote the iteration number and reconstruction PSNR, respectively.

Fig. 3: Phase-transition diagrams of the four factorization methods on synthetic data: (a) WS p MF-II, (b) WS p TF-II, (c) WS p MF-I, and (d) WS p TF-I. The horizontal axis denotes input SNR, the vertical axis denotes sampling ratio, and gray level represents empirical completion probability from $0$ (black) to $1$ (white).

Fig. 4: Visual restoration results of the matrix-based methods on BSD color images. (a) Original image; (b) incomplete and corrupted observation; (c) RPCA; (d) Unifying; (e) GIRNN; (f) RLMF; (g) WS p MF-I (Ours); and (h) WS p MF-II (Ours).

Fig. 5: Visual restoration results of the tensor-based methods on BSD color images. (a) Original image; (b) incomplete and corrupted observation; (c) SNN; (d) TNN; (e) RLTF; (f) WNNTF; (g) SPTF; (h) WTSP; (i) GTNLN; (j) WS p TF-I (Ours); and (k) WS p TF-II (Ours).

Fig. 6: Image-wise PSNR and SSIM for the 15 BSD test images at sampling ratios of

30\%

and

50\%

Fig. 7: Parameter sensitivity of the four proposed models in the color image restoration experiment. The horizontal axes denote $p$ and $q$ , and the vertical axis denotes PSNR (dB). (a) WS p TF-II; (b) WS p MF-II; (c) WS p MF-I; and (d) WS p TF-I. For WS p TF-I and WS p MF-I, only parameter pairs satisfying $q\geq 1-\frac{4p^{2}}{1+4p}$ are retained; for WS p TF-II and WS p MF-II, only pairs satisfying $1\leq p+q\leq 2$ are retained.

Method Type	Method	PSNR (dB)		SSIM		FSIM		Time (s)
Method Type	Method	30% Sampling	50% Sampling	30% Sampling	50% Sampling	30% Sampling	50% Sampling	30% Sampling	50% Sampling
Matrix-Based Methods	RPCA	23.23	26.46	0.81	0.89	0.80	0.87	6.72	7.60
	Unifying	23.14	25.19	0.80	0.84	0.80	0.83	1.55	1.86
	GIRNN	22.53	27.04	0.76	0.87	0.78	0.86	6.51	7.67
	RLMF	21.47	25.68	0.67	0.84	0.74	0.84	1.04	1.60
	WS p MF-I (Ours)	22.76	25.47	0.74	0.84	0.77	0.83	1.21	1.65

TABLE V: Average PSNR, SSIM, FSIM, and running time for color-image restoration at different sampling ratios. For the reconstruction-quality metrics, the best and second-best available results among all methods are marked in bold and underlined , respectively. Lower running times are better.

Fig. 8: Evolution of the active factor width (current rank) of WS p TF-II on the five HSI scenes. All runs are initialized with

r_{0}=180

, and the scene-dependent decreases reflect finite pruning of low-energy column pairs according to ( 37 ).

Fig. 9: Visual comparison of hyperspectral image inpainting results and the corresponding spectral curves on five representative HSI scenes. Columns (a)–(j) correspond to Observed, TNN, SNN, WNNTF, SPTF, WTSP, RLTF, GTNLN, WS p TF-I (Ours), and WS p TF-II (Ours), respectively. For each scene, the upper row shows the restoration result and the lower row shows the corresponding spectral curve. The red and blue boxes indicate representative enlarged spatial regions, while the curves compare the reconstructed and original spectral responses.

Panel of Table 15 — TABLE VI: Quantitative HSI inpainting results at a sampling ratio of $30\%$ under salt-and-pepper noise densities of $0.05$ and $0.10$ . For each scene, metric, and noise density, the best and second-best results are marked in bold and underlined , respectively.

PCB	Method	Missing hole	Mouse bite	Open circuit	Short circuit	Spur	Spurious copper	Average
PCB1	WS p TF-I (Ours)	0.9705	0.8652	0.8298	0.8583	0.7838	0.9603	0.8780
	WS p TF-II (Ours)	0.9750	0.8641	0.9041	0.9135	0.8092	0.9389	0.9008
	WNNTF	0.9690	0.8090	0.8625	0.8850	0.6986	0.9487	0.8621
	SPTF	0.9703	0.8467	0.8351	0.8583	0.7500	0.9594	0.8700
	SNN	0.7656	0.7340	0.7042	0.6843	0.6115	0.7233	0.7038
	TNN	0.8928	0.7012	0.8842	0.8212	0.5691	0.7535	0.7703

TABLE VII: F-measure comparison for PCB defect detection. For each PCB group and defect category, the best and second-best results are marked in bold and underlined , respectively.

Fig. 10: Visual comparison results of different methods on PCB defect detection. The first two rows correspond to the first frontal slice of PCB1, and the last two rows correspond to the tenth frontal slice of PCB2. The first and third rows show the original defective image in column (a) and the recovered backgrounds in columns (b)–(j); the second and fourth rows show the corresponding ground-truth mask in column (a) and predicted defect masks in columns (b)–(j). (a) Original/ground truth; (b) WS p TF-I; (c) WNNTF; (d) SPTF; (e) WS p TF-II; (f) SNN; (g) TNN; (h) RLTF; (i) WTSP; (j) GTNLN.

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