"Samplet basis pursuit: Multiresolution scattered data approximation with sparsity constraints"Multerer, MichaelWe consider scattered data approximation in samplet coordinates with l1-regularization. The application of an l1-regularization term enforces sparsity of the coefficients with respect to the samplet basis. 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. By using the Riesz isometry, we embed samplets into reproducing kernel Hilbert spaces and discuss the properties of the resulting functions. We argue that the class of signals that are sparse with respect to the embedded samplet basis is considerably larger than the class of signals that are sparse with respect to the basis of kernel translates. Vice versa, every signal that is a linear combination of only a few kernel translates is sparse in samplet coordinates. Therefore, samplets enable the use of well-established multiresolution techniques on general scattered data sets. |
« back