We consider scattered data approximation in samplet coordinates with $\ell_{1}$ -regularization. The application of an $\ell_{1}$ -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. Therefore, samplets enable the use of well-established multiresolution techniques on general scattered data sets. 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. We propose the rapid solution of the problem under consideration by combining soft-shrinkage with the semi-smooth Newton method. Leveraging on the sparse representation of kernel matrices in samplet coordinates, this approach converges faster than the fast iterative shrinkage thresholding algorithm and is feasible for large-scale data. Numerical benchmarks are presented and demonstrate the superiority of the multiresolution approach over the single-scale approach. As large-scale applications, the surface reconstruction from scattered data and the reconstruction of scattered temperature data using a dictionary of multiple kernels are considered.

中文翻译：

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样本基础追求：具有稀疏性约束的多分辨率分散数据近似

我们考虑样本坐标中的离散数据近似$\ell_{1}$ -正则化。的应用$\ell_{1}$ -正则化项强制系数相对于样本基础的稀疏性。样本是小波类型的带符号测量，针对分散数据量身定制。因此，样本使得能够在一般分散数据集上使用完善的多分辨率技术。它们在定位、多分辨率分析和数据压缩方面提供与小波类似的特性。通过使用 Riesz 等距，我们将样本嵌入到再现核希尔伯特空间中并讨论所得函数的属性。我们认为，相对于嵌入样本基础稀疏的信号类别比相对于内核转换基础稀疏的信号类别大得多。反之亦然，仅由几个内核平移组成的线性组合的每个信号在样本坐标中都是稀疏的。我们提出通过将软收缩与半平滑牛顿法相结合来快速解决所考虑的问题。利用样本坐标中核矩阵的稀疏表示，该方法比快速迭代收缩阈值算法收敛得更快，并且对于大规模数据是可行的。提出了数值基准并证明了多分辨率方法相对于单尺度方法的优越性。作为大规模应用，考虑了散射数据的表面重建和使用多核字典的散射温度数据的重建。