Approximating a Gram Matrix for Improved Kernel-Based Learning

Abstract

A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O ( n ^3), 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 ${\tilde G}_{k} = CW^{+}_{k}C^{T}$ , where C is a matrix consisting of a small number c of columns of G and W _ k 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 G _ k be the best rank- k approximation to G . We prove that by choosing O ( k / ε ^4) columns ${\left\|G - CW^{+}_{k}C^{T}\right\|_{\xi}} \leq \|G - G_{k}\|_{\xi} + \sum\limits_{i=1}^{n} G^{2}_{ii},$ both in expectation and with high probability, for both ξ = 2, F , and for all k : 0 ≤  k  ≤ rank(W). This approximation can be computed using O ( n ) additional space and time, after making two passes over the data from external storage.