How could I reverse the effect of kernel fisher?

Question

I have used Kernel fisher's discriminant analysis in my project and it worked just great. but my problem arises from the fact that when I mapped my data set using kernel functions, all data and also all eigenvalues and eigenvectors are in that space and for testing new samples I face some problems. let me explain it with an example. when I have for example 50 samples with 10 features for describing each sample, my data matrix is 50 by 10 and mapping this function will result in a 50 by 50 matrix in the new feature space. so the eigenvectors (W in FDA) are also in 50D space. now for testing a new sample that is a vector with 10 elements as its features, the mapped data matrix will be 10 by 10 and it is not in the 50D space, so I can't project it into W to obtain which class does it belong to... pleas help me, what could I do?

Answer 1

You are not supposed to map testing points against themselves but against training set. This is why kernel methods (especially not sparse) in general do not scale well - you have to keep the old training set all the time. Thus you will obtain your projection through K(TEST_SAMPLES, TRAINING_SET) which is 10x50 and can be used in your 50 dimensional space.