We consider kernel-based learning in samplet coordinates with
l1-regularization. The application of an l1-regularization term enforces
sparsity of the coefficients with respect to the samplet basis. Therefore, we
call this approach samplet basis pursuit. Samplets are wavelet-type signed
measures, which are tailored to scattered data. They provide similar properties
as wavelets in terms of localization, multiresolution analysis, and data
compression. The class of signals that can sparsely be represented in a samplet
basis is considerably larger than the class of signals which exhibit a sparse
representation in the single-scale basis. In particular, every signal that can
be represented by the superposition of only a few features of the canonical
feature map is also sparse in samplet coordinates. We propose the efficient
solution of the problem under consideration by combining soft-shrinkage with
the semi-smooth Newton method and compare the approach to the fast iterative
shrinkage thresholding algorithm. We present numerical benchmarks as well as
applications to surface reconstruction from noisy data and to the
reconstruction of temperature data using a dictionary of multiple kernels