Kernel discriminant analysis for positive definite and indefinite kernels

Elzbieta Pekalska, Bernard Haasdonk

    Research output: Contribution to journalArticlepeer-review


    Kernel methods are a class of well established and successful algorithms for pattern analysis thanks to their mathematical elegance and good performance. Numerous nonlinear extensions of pattern recognition techniques have been proposed so far based on the so-called kernel trick. The objective of this paper is twofold. First, we derive an additional kernel tool that is still missing, namely kernel quadratic discriminant (KQD). We discuss different formulations of KQD based on the regularized kernel Mahalanobis distance in both complete and class-related subspaces. Secondly, we propose suitable extensions of kernel linear and quadratic discriminants to indefinite kernels. We provide classifiers that are applicable to kernels defined by any symmetric similarity measure. This is important in practice because problem-suited proximity measures often violate the requirement of positive definiteness. As in the traditional case, KQD can be advantageous for data with unequal class spreads in the kernel-induced spaces, which cannot be well separated by a linear discriminant. We illustrate this on artificial and real data for both positive definite and indefinite kernels. © 2009 IEEE.
    Original languageEnglish
    Pages (from-to)1017-1031
    Number of pages14
    JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
    Issue number6
    Publication statusPublished - 2009


    • Discriminant analysis
    • Indefinite kernels
    • Kernel methods
    • Machine learning
    • Pattern recognition


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