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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.