Approximating a Gram Matrix for Improved Kernel-Based Learning

A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an n × n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form G̃k = CW + k C T , where C is a matrix consisting of a small number c of columns of G and Wk is the best rank-k approximation to W , the matrix formed by the intersection between those c columns of G and the corresponding c rows of G. An important aspect of the algorithm is the probability distribution used to randomly sample the columns; we will use a judiciously-chosen and data-dependent nonuniform probability distribution. Let ‖·‖2 and ‖·‖F denote the spectral norm and the Frobenius norm, respectively, of a matrix, and let Gk be the best rank-k approximation to G. We prove that by choosing O(k/ ) columns ∥ ∥ ∥G− CW k C ∥ ∥ ∥ ξ ≤ ‖G−Gk‖ξ + n ∑