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Recently the kernel discriminant analysis (KDA) has been successfully applied in many applications. However, kernel functions are usually defined a priori and it is not known what the optimum kernel function for nonlinear discriminant analysis is. Otsu derived the optimum nonlinear discriminant analysis (ONDA) by assuming the underlying probabilities similar with the Bayesian decision theory. Kurita derived discriminant kernels function (DKF) as the optimum kernel functions in terms of the discriminant criterion by investigating the optimum discriminant mapping constructed by the ONDA. The derived kernel function is defined by using the Bayesian posterior probabilities. We can define a family of DKFs by changing the estimation method of the Bayesian posterior probabilities. In this paper, we propose a novel discriminant kernel function based on L1-regularized regression, called L1 DKF. L1 DKF is given by using the Bayesian posterior probabilities estimated by L1 regression. Since L1 regression yields a sparse representation for given samples, we can naturally introduce the sparseness into the discriminant kernel function. To introduce the sparseness into LDA, we use L1 DKF as the kernel function of LDA. In experiments, we show sparseness and classification performance of L1 DKF.

Sparse Discriminant Analysis Based on the Bayesian Posterior Probability Obtained by L1 Regression