Low-Rank Matrix Recovery from Noise via an MDL Framework-based Atomic Norm

TL;DR

This paper introduces a novel low-rank matrix recovery method using an MDL framework and atomic norm, effectively handling unknown rank and sparse outliers in noisy data, outperforming existing methods especially with limited observations or high corruption.

Contribution

The study proposes a new approach combining MDL and atomic norm for robust low-rank matrix recovery without prior knowledge of rank or outlier locations, improving success rates.

Findings

Higher success rate than state-of-the-art methods

Effective with limited observations and high corruption

Demonstrated robustness on synthetic and real data

Abstract

The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure and the particular locations and values of the sparse outliers are not known. Thus, the conventional methods cannot separate the low-rank and sparse components completely, especially in the case of gross outliers or deficient observations. Therefore, in this study, we employ the minimum description length (MDL) principle and atomic norm for low-rank matrix recovery to overcome these limitations. First, we employ the atomic norm to find all the candidate atoms of low-rank and sparse terms, and then we minimize the description length of the model in order to select the appropriate atoms of low-rank and the sparse matrices, respectively. Our experimental analyses show that the proposed approach can obtain a higher success rate than the state-of-the-art methods, even when the number of observations is limited or the corruption ratio is high. Experimental results utilizing synthetic data and real sensing applications (high dynamic range imaging, background modeling, removing noise and shadows) demonstrate the effectiveness, robustness and efficiency of the proposed method.