Data representations and generalization error in kernel based learning machines

This paper focuses on the problem of how data representation influences the generalization error of kernel based learning machines like Support Vector Machines (SVM) for classification. Frame theory provides a well founded mathematical framework for representing data in many different ways. We analyze the effects of sparse and dense data representations on the generalization error of such learning machines measured by using leave-one-out error given a finite amount of training data. We show that, in the case of sparse data representations, the generalization error of an SVM trained by using polynomial or Gaussian kernel functions is equal to the one of a linear SVM. This is equivalent to saying that the capacity of separating points of functions belonging to hypothesis spaces induced by polynomial or Gaussian kernel functions reduces to the capacity of a separating hyperplane in the input space. Moreover we show that, in general, sparse data representations increase or leave unchanged the generalization error of kernel based methods. %as long as the representation is not too sparse, as in the case of %very large dictionaries. Very sparse representations increase %drastically the generalization error of kernel based methods. Dense data representations, on the contrary, reduce the generalization error in the case of very large frames. We use two different schemes for representing data in overcomplete systems of Haar and Gabor functions, and measure SVM generalization error on benchmarked data sets.