Robust Nonnegative Matrix Factorization via $L_1$ Norm Regularization

TL;DR

This paper introduces a Robust Nonnegative Matrix Factorization method that effectively handles unknown-position large additive noise, improving data approximation and outlier detection in applications like face recognition.

Contribution

It proposes a novel NMF algorithm that models and estimates outliers without prior noise position information, with a solid theoretical basis and efficient optimization.

Findings

Outperforms existing NMF variants in noisy data scenarios

Accurately detects outliers and noise positions

Enhances data reconstruction quality

Abstract

Nonnegative Matrix Factorization (NMF) is a widely used technique in many applications such as face recognition, motion segmentation, etc. It approximates the nonnegative data in an original high dimensional space with a linear representation in a low dimensional space by using the product of two nonnegative matrices. In many applications data are often partially corrupted with large additive noise. When the positions of noise are known, some existing variants of NMF can be applied by treating these corrupted entries as missing values. However, the positions are often unknown in many real world applications, which prevents the usage of traditional NMF or other existing variants of NMF. This paper proposes a Robust Nonnegative Matrix Factorization (RobustNMF) algorithm that explicitly models the partial corruption as large additive noise without requiring the information of positions of noise. In practice, large additive noise can be used to model outliers. In particular, the proposed method jointly approximates the clean data matrix with the product of two nonnegative matrices and estimates the positions and values of outliers/noise. An efficient iterative optimization algorithm with a solid theoretical justification has been proposed to learn the desired matrix factorization. Experimental results demonstrate the advantages of the proposed algorithm.