Graph Regularized Sparse $L_{2,1}$ Semi-Nonnegative Matrix Factorization for Data Reduction

TL;DR

This paper introduces a robust $L_{2,1}$ semi-nonnegative matrix factorization algorithm that enhances data reduction by effectively handling noise and outliers, with proven convergence and superior performance on benchmark datasets.

Contribution

The paper proposes a novel $L_{2,1}$ SNF algorithm that improves stability and noise resistance over traditional SNF methods, with theoretical convergence analysis.

Findings

Outperforms conventional SNF under Gaussian noise

Demonstrates stability and robustness on benchmark datasets

Provides theoretical proof of monotonic convergence

Abstract

Non-negative Matrix Factorization (NMF) is an effective algorithm for multivariate data analysis, including applications to feature selection, pattern recognition, and computer vision. Its variant, Semi-Nonnegative Matrix Factorization (SNF), extends the ability of NMF to render parts-based data representations to include mixed-sign data. Graph Regularized SNF builds upon this paradigm by adding a graph regularization term to preserve the local geometrical structure of the data space. Despite their successes, SNF-related algorithms to date still suffer from instability caused by the Frobenius norm due to the effects of outliers and noise. In this paper, we present a new $L_{2,1}$ SNF algorithm that utilizes the noise-insensitive $L_{2,1}$ norm. We provide monotonic convergence analysis of the $L_{2,1}$ SNF algorithm. In addition, we conduct numerical experiments on three benchmark mixed-sign datasets as well as several randomized mixed-sign matrices to demonstrate the performance superiority of $L_{2,1}$ SNF over conventional SNF algorithms under the influence of Gaussian noise at different levels.