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.

翻译：非负矩阵分解（NMF）是一种有效的多元数据分析算法，广泛应用于特征选择、模式识别和计算机视觉等领域。其变体——半非负矩阵分解（SNF）——扩展了NMF生成基于部件的数据表示的能力，使其能够处理混合符号数据。图正则化SNF在此基础上，通过引入图正则项来保持数据空间的局部几何结构。尽管取得了成功，但现有的SNF相关算法仍因Frobenius范数对离群点和噪声敏感而存在不稳定性。本文提出了一种新的$L_{2,1}$ SNF算法，该算法采用对噪声不敏感的$L_{2,1}$范数。我们给出了$L_{2,1}$ SNF算法的单调收敛性分析。此外，我们在三个基准混合符号数据集以及多个随机生成的混合符号矩阵上进行了数值实验，结果表明在不同强度的高斯噪声影响下，$L_{2,1}$ SNF算法性能均优于传统SNF算法。