Multiscale models are known to be successful in uncovering and representing structure in data at different resolutions. We propose here a feature driven Reproducing Kernel Hilbert Space (RKHS) for which the associated kernel has a weighted multiscale structure. For generating approximations in this space, we provide a practical forward–backward algorithm that is shown to greedily construct a set of basis functions having a multiscale structure which enables sparse efficient representation of the given data and efficient predictions. We provide a detailed analysis of the algorithm including recommendations for selecting algorithmic hyperparameters and estimating probabilistic rates of convergence at individual scales. We also extend this analysis to a multiscale setting, studying the effects of finite scale truncation and quality of solution in the inherent RKHS. In the last section, we analyze the performance of the approach on a variety of simulations and real data sets illustrating the efficiency claims in terms of model quality and data reduction.

众所周知，多尺度模型可以成功地揭示和表示不同分辨率的数据结构。我们在这里提出了一个特征驱动的再生内核希尔伯特空间（RKHS），其相关的内核具有加权多尺度结构。为了在该空间中生成近似值，我们提供了一种实用的前向后向算法，该算法被证明可以贪婪地构造一组具有多尺度结构的基函数，从而能够对给定数据进行稀疏有效的表示并进行有效的预测。我们提供了算法的详细分析，包括选择算法超参数和估计单个尺度的收敛概率率的建议。我们还将这种分析扩展到多尺度设置，研究固有 RKHS 中有限尺度截断和解质量的影响。在最后一节中，我们分析了该方法在各种模拟和真实数据集上的性能，说明了模型质量和数据缩减方面的效率要求。