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Classical linear discriminant analysis (LDA) is based on squared Frobenious norm and hence is sensitive to outliers and noise. To improve the robustness of LDA, in this paper, we introduce capped l2,1-norm of a matrix, which employs nonsquared l2-norm and "capped" operation, and further propose a novel capped l2,1-norm linear discriminant analysis, called CLDA. Due to the use of capped l2,1-norm, CLDA can effectively remove extreme outliers and suppress the effect of noise data. In fact, CLDA can be also viewed as a weighted LDA. CLDA is solved through a series of generalized eigenvalue problems with theoretical convergency. The experimental results on an artificial data set, some UCI data sets and two image data sets demonstrate the effectiveness of CLDA.

Compared to the standard support vector machine, the generalized eigenvalue proximal support vector machine coped well with the “Xor” problem. However, it was based on the squared Frobenius norm and hence was sensitive to outliers and noise. To improve the robustness, this paper introduces capped L 1 -norm into the generalized eigenvalue proximal support vector machine, which employs nonsquared L 1 -norm and “capped” operation, and further proposes a novel capped L 1 -norm proximal support vector machine, called CPSVM. Due to the use of capped L 1 -norm, CPSVM can effectively remove extreme outliers and suppress the effect of noise data. CPSVM can also be viewed as a weighted generalized eigenvalue proximal support vector machine and is solved through a series of generalized eigenvalue problems. The experimental results on an artificial dataset, some UCI datasets, and an image dataset demonstrate the effectiveness of CPSVM.