Using Multiscale Kernel Models to Reconstruct Multivariate Functions from Scattered Data

This paper reconstructs multivariate functions from scattered data by a new multiscale technique. The reconstruction uses support vector regression model by positive definite reproducing kernels in Hilbert spaces. But it adopts techniques from wavelet theory and shift-invariant spaces to construct a new class of kernels as multiscale superpositions of shifts and scales of a single compactly supported function. This means that the advantages of scaled regular grids are used to construct the kernels, while the advantages of unrestricted scattered data interpolation are maintained after the kernels are constructed. Using such a multiscale kernel, the reconstruction method interpolates at given scattered data. No manipulations of the data are needed. Then, the multiscale structure of the kernel allows to represent the interpolant on regular grids on all scales involved, with cheap evaluation due to the compact support of the function, and with a recursive evaluation technique in support vector regression. Numerical comparisons are given in two dimensions which show competitive results with the single-variable two-variable function, the new model not only can reconstruct linear and the non-linear combination functions very well, but also performs better in multivariate functions reconstruction. The results indicate that the proposed method has effectiveness in terms of both objective measurements and visual evaluation.