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

TL;DR

This paper proposes a modified nuclear norm framework that captures both local and global low-rank structures in matrix recovery, providing theoretical guarantees and improved performance over existing methods.

Contribution

Introduction of a flexible modified nuclear norm framework that guarantees exact recovery in matrix completion and Robust PCA without parameter tuning.

Findings

The MNN framework guarantees exact recovery under mild assumptions.

MNN captures both local and global information without trade-off parameters.

Experimental results show superior performance of MNN over existing methods.

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.