Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm

2025-07-24

AI Summary

Nuclear norm minimization in matrix recovery struggles to simultaneously preserve local structural information and enforce global low-rankness, while its theoretical guarantees critically depend on careful hyperparameter tuning. Method: This paper proposes the Modified Nuclear Norm (MNN) framework, which integrates local features into low-rank modeling via learnable or predefined matrix transformations—enabling joint local-global representation without explicit trade-off parameters. Contribution/Results: MNN provides the first provable exact recovery guarantees for robust PCA and matrix completion without introducing balancing parameters. It is universally compatible with diverse structural transformations, forming a unified framework for structured low-rank recovery. Extensive experiments demonstrate that MNN consistently outperforms state-of-the-art methods across multiple benchmark tasks. The implementation, along with supplementary materials, is publicly released, confirming its effectiveness and generalizability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Enhancing matrix recovery by capturing local and global low-rank structures
• Providing exact recovery guarantees for Robust PCA and matrix completion
• Unifying structured low-rank recovery with flexible transformation options

Innovation

• Modified Nuclear Norm captures local and global information
• MNN provides exact theoretical recovery guarantees
• Flexible design accommodates various transformations