Sparse Greedy Matrix Approximation for Machine Learning

Abstract

In kernel based methods such as Regularization Networks large datasets pose signi- cant problems since the number of basis functions required for an optimal solution equals the number of samples. We present a sparse greedy approximation technique to construct a compressed representation of the design matrix. Experimental results are given and connections to Kernel-PCA, Sparse Kernel Feature Analysis, and Matching Pursuit are pointed out. 1. Introduction Many recent advances in machine learning such as Support Vector Machines [Vapnik, 1995], Regularization Networks [Girosi et al., 1995], or Gaussian Processes [Williams, 1998] are based on kernel methods. Given an m-sample f(x 1 ; y 1 ); : : : ; (x m ; y m )g of patterns x i 2 X and target values y i 2 Y these algorithms minimize the regularized risk functional min f2H R reg [f ] = 1 m m X i=1 c(x i ; y i ; f(x i )) + 2 kfk 2 H : (1) Here H denotes a reproducing kernel Hilbert space (RKHS) [Aronszajn, 1950],...