Fisher first introduced the Fisher linear discriminant back in 1938. After the popularization of the support vector machine (SVM) and the kernel trick it became inevitable that the Fisher linear discriminant would be kernelized. Sebastian Mika accomplished this task as part of his Ph.D. in 2002 and the kernelized Fisher discriminant (KFD) now forms part of the largescale machine-learning tool Shogun. In this article we introduce the package MathKFD. We apply MathKFD to synthetic datasets to demonstrate nonlinear classification via kernels. We also test performance on datasets from the machine-learning literature. The construction of MathKFD follows closely in style the construction of MathSVM by Nilsson and colleagues. We hope these two packages and others of the same ilk will eventually be integrated to form a kernel-based machine-learning environment for Mathematica. ‡ Introduction A two-class machine-learning problem requires learning how to discriminate between data points x i in sample space X belonging to classes y i oe 8+1, -1<, when given only a set of examples 8x i , y i < from each class. The currently popular support vector machine [1] solves this problem through the construction of a hyperplane w.x + b that separates the data points x i , in the sense that all the x i of a given class are on the same side of the plane. In SVMs the separating plane is chosen to maximize the distances from it to the closest data points.The original multidimensional machine-learning algorithm [2] solves the same problem by maximizing between-class to within-class scatter ratio. In this article we describe Fisher's technique and how the introduction of a kernel allows nonlinear classifiers. We build a Kernelized Fisher Discriminant package, MathKFD, and explore its classification capabilities using synthetic and real datasets.The Mathematica Journal 13

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