setting equal to zero gives Since only the direction of , and hence the direction of matters, the above can be solved for as Note that in practice, is usually singular and so a multiple of the identity is added to it Given the solution for , the projection of a new data point is given by Multi-class KFD The extension to cases where there are more than two classes is relatively straightforward. Let be the number of classes. Then multi-class KFD involves projecting the data into a -dimensional space using discriminant functions This can be written in matrix notation where the are the columns of . Further, the between-class covariance matrix is now where is the mean of all the data in the new feature space. The within-class covariance matrix is The solution is now obtained by maximizing The kernel trick can again be used and the goal of multi-class KFD becomes where and The are defined as in the above section and is defined as can then be computed by finding the leading eigenvectors of . Furthermore, the projection of a new input, , is given by where the component of is given by . Classification using KFD In both two-class and multi-class KFD, the class label of a new input can be assigned as where is the projected mean for class and is a distance function. Applications Kernel discriminant analysis has been used in a variety of applications. These include: Face recognition and detection Hand-written digit recognition Palmprint recognition Classification of malignant and benign cluster microcalcifications Seed classification Search for the Higgs Boson at CERN See also Factor analysis Kernel principal component analysis Kernel trick Linear discriminant analysis References External links Kernel Discriminant Analysis in C# - C# code to perform KFD. Matlab Toolbox for Dimensionality Reduction - Includes a method for performing KFD. Handwriting Recognition using Kernel Discriminant Analysis - C# code that demonstrates handwritten digit recognition using KFD. Statistical classification