Approximating a gram matrix for improved kernel-based learning (Extended 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 X n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form G(k) = (CWk+CT), 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 parallel to (.) parallel to and parallel to (.) parallel to(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/epsilon(4)) columns parallel to G-(CWk+CT)parallel to(xi) <= parallel to(xi) <= parallel to G-G(k)parallel to(xi) + epsilon (i=1)Sigma(n) G(ii)(2), both in expectation and with high probability, for both xi = 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.