提出新型核范数框架，同时捕捉局部与全局低秩结构。
Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm
- 通过变换矩阵后应用核范数，融合局部与全局信息。
- 在鲁棒PCA和矩阵补全任务中实现理论可证明的精确恢复。
- 无需调参，适合需高精度恢复的结构化低秩数据场景。
核范数（NN）广泛应用于矩阵恢复问题，如鲁棒主成分分析（Robust PCA）和矩阵补全（MC），利用数据固有的全局低秩特性。本文提出一种新的改进核范数（MNN）框架，其通过适当的矩阵变换并在此基础上应用核范数定义一族新范数。MNN框架具有两大优势：（1）无需调节权衡参数即可联合捕捉局部信息与全局低秩性；（2）在变换满足温和假设条件下，为鲁棒PCA和矩阵补全任务提供了精确的理论恢复保证——这是现有结合局部与全局信息的方法所不具备的。得益于其通用且灵活的设计，MNN可兼容多种已验证的变换，提供统一高效的结构化低秩恢复方法。大量实验验证了该方法的有效性。代码与补充材料见 https://github.com/andrew-pengjj/modified_nuclear_norm。
原文摘要 · Abstract (English)
The nuclear norm (NN) has been widely explored in matrix recovery problems, such as Robust PCA and matrix completion, leveraging the inherent global low-rank structure of the data. In this study, we introduce a new modified nuclear norm (MNN) framework, where the MNN family norms are defined by adopting suitable transformations and performing the NN on the transformed matrix. The MNN framework offers two main advantages: (1) it jointly captures both local information and global low-rankness without requiring trade-off parameter tuning; (2) Under mild assumptions on the transformation, we provided exact theoretical recovery guarantees for both Robust PCA and MC tasks-an achievement not shared by existing methods that combine local and global information. Thanks to its general and flexible design, MNN can accommodate various proven transformations, enabling a unified and effective approach to structured low-rank recovery. Extensive experiments demonstrate the effectiveness of our method. Code and supplementary material are available at https://github.com/andrew-pengjj/modified_nuclear_norm.
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