## Multidimensional smoothing using hyperbolic interpolatory wavelets

Markus Hegland, Ole M. Nielsen, and Zuowei Shen

### Abstract

We propose the application of hyperbolic interpolatory wavelets for large-scale $d$-dimensional data fitting. In particular, we show how wavelets can be used as a highly efficient tool for multidimensional smoothing. The grid underlying these wavelets is a sparse grid. The hyperbolic interpolatory wavelet space of level $j$ uses $O(j^{d-1}2^j)$ basis functions and it is shown that under sufficient smoothness an approximation error of order $O\left(\left(\begin{array}{c}j+d-1 \ d-1\end{array}\right) 2^{-2j}\right)$ can be achieved. The implementation uses the fast wavelet transform and an efficient indexing method to access the wavelet coefficients. A practical example demonstrates the efficiency of the approach.

Full Text (PDF) [182 KB]

### Key words

sparse grids, predictive modelling, wavelets, smoothing, data mining.

### AMS subject classifications

65C60, 65D10, 65T60.

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