Samplets: Construction and scattered data compression

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively, is $O(N)$, where $N$ is the number of data points. Samplets with vanishing moments can be used to compress kernel matrices, arising, for instance, kernel based learning and scattered data approximation. The result are sparse matrices with only $O(N \log N )$ remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computational cost $O(N \log N )$. The accuracy of the approximation is controlled by the number of vanishing moments. Besides the cost efficient storage of kernel matrices, the sparse representation enables the application of sparse direct solvers for the numerical solution of corresponding linear systems. In addition to a comprehensive introduction to samplets and their properties, we present numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large scattered data sets accessible for multiresolution analysis.