  Data sites selected from modeling high-dimensional problems often appear
scattered in non-paternalistic ways. Except for sporadic clustering at some
spots, they become relatively far apart as the dimension of the ambient space
grows. These features defy any theoretical treatment that requires local or
global quasi-uniformity of distribution of data sites. Incorporating a
recently-developed application of integral operator theory in machine learning,
we propose and study in the current article a new framework to analyze kernel
interpolation of high dimensional data, which features bounding stochastic
approximation error by the spectrum of the underlying kernel matrix. Both
theoretical analysis and numerical simulations show that spectra of kernel
matrices are reliable and stable barometers for gauging the performance of
kernel-interpolation methods for high dimensional data.
