Linear Discriminant Analysis Based on L1-Norm Maximization

TLDR

Linear discriminant analysis (LDA) is a widely used dimensionality reduction technique, but its L2‑norm objective makes it sensitive to outliers. This work introduces a robust LDA variant that maximizes the ratio of L1‑norm based between‑class to within‑class dispersion to obtain local optimal projection vectors. The method is theoretically proven to be feasible, robust to outliers, and to avoid the singular within‑class scatter matrix problem of conventional LDA. Experiments on synthetic, benchmark classification, and three image datasets confirm the method’s effectiveness.

Abstract

Linear discriminant analysis (LDA) is a well-known dimensionality reduction technique, which is widely used for many purposes. However, conventional LDA is sensitive to outliers because its objective function is based on the distance criterion using L2-norm. This paper proposes a simple but effective robust LDA version based on L1-norm maximization, which learns a set of local optimal projection vectors by maximizing the ratio of the L1-norm-based between-class dispersion and the L1-norm-based within-class dispersion. The proposed method is theoretically proved to be feasible and robust to outliers while overcoming the singular problem of the within-class scatter matrix for conventional LDA. Experiments on artificial datasets, standard classification datasets and three popular image databases demonstrate the efficacy of the proposed method.