Improved Bound for the Nystrom's Method and its Application to Kernel Classification

Abstract: We develop two approaches for analyzing the approximation error bound for the Nystr\"{o}m method, one based on the concentration inequality of integral operator, and one based on the compressive sensing theory. We show that the approximation error, measured in the spectral norm, can be improved from $O(N/\sqrt{m})$ 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 $\rho \in (0, 1/2)$ is a positive constant that characterizes the eigengap. When the eigenvalues of the kernel matrix follow a $p$-power law, our analysis based on compressive sensing theory further improves the bound to $O(N/m^{p - 1})$ under an incoherence assumption, which explains why the Nystr\"{o}m method works well for kernel matrix with skewed eigenvalues. We present a kernel classification approach based on the Nystr\"{o}m method and derive its generalization performance using the improved bound. We show that when the eigenvalues of kernel matrix follow a $p$-power law, we can reduce the number of support vectors to $N^{2p/(p^2 - 1)}$, a number less than $N$ when $p > 1+\sqrt{2}$, without seriously sacrificing its generalization performance.