Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm
This addresses matrix recovery for applications in data analysis and machine learning, offering a novel unified approach with proven guarantees.
The paper tackles matrix recovery problems like Robust PCA and matrix completion by introducing a modified nuclear norm (MNN) framework that captures both local information and global low-rankness without parameter tuning, providing theoretical recovery guarantees and demonstrating effectiveness in experiments.
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.