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.

翻译：我们考虑在采样基坐标下结合l1正则化的核方法学习。l1正则化项的应用强制了关于采样基系数的稀疏性，因此我们将此方法称为采样基追踪。采样基是针对散乱数据量身定制的类小波符号测度，在局部化、多分辨率分析和数据压缩等方面具有与小波相似的性质。可在采样基下稀疏表示的信号类别，远大于可在单尺度基下稀疏表示的信号类别。特别地，每个可通过规范特征映射的少数特征叠加来表示的信号，在采样基坐标下同样是稀疏的。我们通过将软阈值法与半光滑牛顿法相结合，提出了该问题的有效求解方案，并将其与快速迭代收缩阈值算法进行了比较。我们给出了数值基准测试，以及从含噪数据中重建曲面和利用多核字典重建温度数据的应用实例。