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We consider scattered data approximation in samplet coordinates with$\ell_1$-regularization. The application of an $\ell_1$-regularization termenforces sparsity of the coefficients with respect to the samplet basis.Samplets are wavelet-type signed measures, which are tailored to scattereddata. Therefore, samplets enable the use of well-established multiresolutiontechniques on general scattered data sets. They provide similar properties aswavelets in terms of localization, multiresolution analysis, and datacompression. By using the Riesz isometry, we embed samplets into reproducingkernel Hilbert spaces and discuss the properties of the resulting functions. Weargue that the class of signals that are sparse with respect to the embeddedsamplet basis is considerably larger than the class of signals that are sparsewith respect to the basis of kernel translates. Vice versa, every signal thatis a linear combination of only a few kernel translates is sparse in sampletcoordinates.  We propose the rapid solution of the problem under consideration by combiningsoft-shrinkage with the semi-smooth Newton method. Leveraging on the sparserepresentation of kernel matrices in samplet coordinates, this approachconverges faster than the fast iterative shrinkage thresholding algorithm andis feasible for large-scale data. Numerical benchmarks are presented anddemonstrate the superiority of the multiresolution approach over thesingle-scale approach. As large-scale applications, the surface reconstructionfrom scattered data and the reconstruction of scattered temperature data usinga dictionary of multiple kernels are considered.

Samplet basis pursuit: Multiresolution scattered data approximation with sparsity constraints