  We develop an improved bound for the approximation error of the Nystr\"{o}m
method under the assumption that there is a large eigengap in the spectrum of
kernel matrix. This is based on the empirical observation that the eigengap has
a significant impact on the approximation error of the Nystr\"{o}m method. Our
approach is based on the concentration inequality of integral operator and the
theory of matrix perturbation. Our analysis shows that when there is a large
eigengap, we can improve the approximation error of the Nystr\"{o}m method from
$O(N/m^{1/4})$ to $O(N/m^{1/2})$ when measured in Frobenius norm, where $N$ is
the size of the kernel matrix, and $m$ is the number of sampled columns.
