Kernel Interpolation of High Dimensional Scattered Data
This addresses the problem of analyzing kernel methods for high-dimensional data in machine learning, but it appears incremental as it builds on existing integral operator theory.
The paper tackles the challenge of kernel interpolation for high-dimensional scattered data, which lacks uniform distribution, by proposing a framework that bounds approximation error using the spectrum of the kernel matrix, with theoretical and numerical results showing spectra as reliable indicators of performance.
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.