Improved Bounds for the Nystrom Method With Application to Kernel Classification

We develop two approaches for analyzing the approximation error bound for the Nystrom method that approximates a positive semidefinite (PSD) matrix by sampling a small set of columns, one based on a concentration inequality for integral operators, and one based on random matrix theory. We show that the approximation error, measured in the spectral norm, can be improved from O(N/root m) to O(N/m(1-rho)) in the case of large eigengap, where N is the total number of data points, m is the number of sampled data points, and rho is an element of (0, 1/2) is a positive constant that characterizes the eigengap. When the eigenvalues of the kernel matrix follow a p-power law, our analysis based on random matrix theory further improves the bound to O(N/m(p-1))under an incoherence assumption. We present a kernel classification approach based on the Nystrom method and derive its generalization performance using the improved bound. We show that when the eigenvalues of the kernel matrix follow a p-power law, we can reduce the number of support vectors to N-2p/(p(2)-1), which is sublinear in N when p > 1 + root 2, without seriously sacrificing its generalization performance.