[论文解读] A Comparative Study for the Weighted Nuclear Norm Minimization and Nuclear Norm Minimization

Zhiyuan Zha, Xinggan Zhang+3 more|arXiv (Cornell University)|Aug 16, 2016

Advanced Image Fusion Techniques被引用 3

一句话总结

本文通过将加权核范数最小化（WNNM）与组稀疏表示（GSR）关联，为WNNM在低秩矩阵逼近中优于核范数最小化（NNM）的现象提供了理论解释。研究表明，WNNM等价于GSR中的加权L1-范数最小化，该方法增强了稀疏性，从而实现更优的秩估计，实验验证表明其在图像去噪性能上优于NNM及当前最先进方法。

ABSTRACT

The nuclear norm minimization (NNM) is commonly used to approximate the matrix rank by shrinking all singular values equally. However, the singular values have clear physical meanings in many practical problems, and NNM may not be able to faithfully approximate the matrix rank. To alleviate the above-mentioned limitation of NNM, recent studies have suggested that the weighted nuclear norm minimization (WNNM) can achieve a better rank estimation than NNM, which heuristically set the weight being inverse to the singular values. However, it still lacks a rigorous explanation why WNNM is more effective than NMM in various applications. In this paper, we analyze NNM and WNNM from the perspective of group sparse representation (GSR). Concretely, an adaptive dictionary learning method is devised to connect the rank minimization and GSR models. Based on the proposed dictionary, we prove that NNM and WNNM are equivalent to L1-norm minimization and the weighted L1-norm minimization in GSR, respectively. Inspired by enhancing sparsity of the weighted L1-norm minimization in comparison with L1-norm minimization in sparse representation, we thus explain that WNNM is more effective than NMM. By integrating the image nonlocal self-similarity (NSS) prior with the WNNM model, we then apply it to solve the image denoising problem. Experimental results demonstrate that WNNM is more effective than NNM and outperforms several state-of-the-art methods in both objective and perceptual quality.

研究动机与目标

• 解决核范数最小化（NNM）在忠实逼近矩阵秩时因对所有奇异值进行等量收缩而导致的局限性。
• 为加权核范数最小化（WNNM）在实践中优于NNM的实证成功提供严谨的理论解释。
• 通过自适应字典学习方法，建立低秩矩阵恢复与组稀疏表示（GSR）之间的联系。
• 展示将非局部自相似性（NSS）与WNNM结合在图像去噪中的有效性。

提出的方法

• 提出一种自适应字典学习方法，将低秩矩阵恢复映射到组稀疏表示（GSR）框架中。
• 证明NNM在GSR模型中等价于L1-范数最小化，而WNNM则对应于加权L1-范数最小化。
• 利用理论等价性解释为何WNNM能增强稀疏性并实现优于NNM的秩逼近。
• 将非局部自相似性（NSS）先验整合到WNNM模型中，以提升图像去噪性能。
• 将图像去噪问题建模为带有NSS约束的WNNM优化问题。
• 采用高效算法求解所得优化问题，并通过大量实验进行验证。

实验结果

研究问题

• RQ1 为何在实践中加权核范数最小化（WNNM）的秩估计优于核范数最小化（NNM）？
• RQ2 低秩矩阵恢复与组稀疏表示（GSR）之间存在何种理论关系？
• RQ3 加权L1-范数最小化中稀疏性的增强如何解释WNNM优于NNM的原因？
• RQ4 将WNNM与非局部自相似性（NSS）结合，能在多大程度上提升图像去噪性能？
• RQ5 所提出的基于WNNM的方法在客观质量与主观感知质量方面，与当前最先进图像去噪技术相比表现如何？

主要发现

• WNNM在组稀疏表示（GSR）框架中理论上等价于加权L1-范数最小化，而NNM则对应于标准L1-范数最小化。
• 加权L1-范数最小化所增强的稀疏性，解释了为何WNNM能提供优于NNM的秩逼近。
• 将非局部自相似性（NSS）与WNNM结合，可显著提升图像去噪性能。
• 所提出的方法在定量指标与主观感知质量方面，均优于多种当前最先进图像去噪方法。
• 实验结果证实，WNNM在各类基准数据集上均显著优于NNM的去噪效果。