Sparse representations and performances in support vector 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 number of training data. We show that, in the case of sparse data representation, the generalization capacity 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. We show that sparse data representations reduce the generalization error as long as the representation is not too sparse, as in the case of very large dictionaries. Dense data representations, on the contrary, reduce the generalization error also in the case of very large dictionaries. We use two different schemes for representing data in overcomplete Haar and Gabor dictionaries, and measure SVM generalization error on bench mark data set. Moreover we study sparse and dense data representations with frame of data and we show how this leads to Principal Component Analysis.