# Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction

arXiv: https://arxiv.org/abs/1512.01003 · HTML: https://mlsanity.com/paper/1512.01003/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2015.

## Built on by (most cited first, 29 of 29)

- [Image Processing and Machine Learning for Hyperspectral Unmixing: An Overview and the HySUPP Python Package](https://mlsanity.com/paper/2308.09375/) (2023)
- [RPCANet++: Deep Interpretable Robust PCA for Sparse Object Segmentation](https://mlsanity.com/paper/2508.04190/) (2025)
- [Truncated tensor Schatten p-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns](https://mlsanity.com/paper/2205.09390/) (2022)
- [Learning Sparse and Low-Rank Priors for Image Recovery via Iterative Reweighted Least Squares Minimization](https://mlsanity.com/paper/2304.10536/) (2023)
- [PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descent](https://mlsanity.com/paper/2204.13940/) (2022)
- [Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction](https://mlsanity.com/paper/2409.07797/) (2024)
- [Robust Implementation of Foreground Extraction and Vessel Segmentation for X-ray Coronary Angiography Image Sequence](https://mlsanity.com/paper/2209.07237/) (2022)
- [Multi-channel Nuclear Norm Minus Frobenius Norm Minimization for Color Image Denoising](https://mlsanity.com/paper/2209.08094/) (2022)
- [Non-Convex Tensor Low-Rank Approximation for Infrared Small Target Detection](https://mlsanity.com/paper/2105.14974/) (2021)
- [Label Learning Method Based on Tensor Projection](https://mlsanity.com/paper/2402.16544/) (2024)
- [Denoising single images by feature ensemble revisited](https://mlsanity.com/paper/2207.05176/) (2022)
- [Multi-View Clustering via Semi-non-negative Tensor Factorization](https://mlsanity.com/paper/2303.16748/) (2023)
- [Hyper-Laplacian Regularized Concept Factorization in Low-rank Tensor Space for Multi-view Clustering](https://mlsanity.com/paper/2304.11435/) (2023)
- [Iterative Reweighted Least Squares Networks With Convergence Guarantees for Solving Inverse Imaging Problems](https://mlsanity.com/paper/2308.05745/) (2023)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Adaptive incomplete multi-view learning via tensor graph completion](https://mlsanity.com/paper/2208.03710/) (2022)
- [Region-wise matching for image inpainting based on adaptive weighted low-rank decomposition](https://mlsanity.com/paper/2303.12421/) (2023)
- [A Novel Truncated Norm Regularization Method for Multi-channel Color Image Denoising](https://mlsanity.com/paper/2307.07932/) (2023)
- [A Theoretically Guaranteed Quaternion Weighted Schatten p-norm Minimization Method for Color Image Restoration](https://mlsanity.com/paper/2307.12656/) (2023)
- [L2,1-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition](https://mlsanity.com/paper/2309.03764/) (2023)
- [Infrared Small Target Detection Using Double-Weighted Multi-Granularity Patch Tensor Model With Tensor-Train Decomposition](https://mlsanity.com/paper/2310.05347/) (2023)
- [One-Step Multi-View Clustering Based on Transition Probability](https://mlsanity.com/paper/2403.01460/) (2024)
- [Interpretable Multi-View Clustering Based on Anchor Graph Tensor Factorization](https://mlsanity.com/paper/2404.00883/) (2024)
- [Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation](https://mlsanity.com/paper/2407.08517/) (2024)
- [Depth-Aided Color Image Inpainting in Quaternion Domain](https://mlsanity.com/paper/2503.16818/) (2025)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [Matrix Completion Via Reweighted Logarithmic Norm Minimization](https://mlsanity.com/paper/2512.21050/) (2025)
- [Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems](https://mlsanity.com/paper/2603.00333/) (2026)

## Abstract (arXiv)

Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten p-Norm Minimization (WSNM), to generalize the NNM to the Schatten p-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex lp-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model complex and dynamic scenes compared with state-of-the-art methods.