This chapter presents strategies for selecting representative node subsets from large datasets while maintaining approximation accuracy. Among these, the Efficient Reduced Basis Algorithm (ERBA) iteratively removes nodes based on residual or power function criteria. Unlike greedy methods that add nodes progressively, ERBA starts with the full dataset and systematically reduces it. A crucial aspect is exploiting the Extended Rippa’s Algorithm to compute validation errors efficiently, Numerical experiments demonstrate that ERBA produces sparse interpolation sets achieving comparable accuracy to full datasets with significant computational savings.

Variably Scaled Kernels (VSKs) replace the global shape parameter with a scaling function, which maps data to an augmented feature space. This construction preserves positive definiteness while increasing separation distances and potentially improving stability. We extend VSKs to discontinuous scaling functions (VSDKs) for approximating functions with jumps, effectively mitigating the Gibbs phenomenon. We establish that VSDKs are limits of continuous VSKs and prove error bounds in piecewise Sobolev spaces, which allow us to show convergence rates that depend only on smoothness within subdomains, not on jump discontinuities.

Optimal scaling function selection critically affects VSK approximation quality. We derive error bounds involving Lebesgue functions, proving that scaling functions mimicking the target reduce pointwise error. This theoretical insight motivates learning the scaling function via discontinuous neural networks (δNNs). The δNN-VSK method jointly optimizes the scaling function and interpolation coefficients by minimizing interpolation error, while the VSK-f approach directly approximates the target. Numerical experiments confirm learned scaling functions closely resemble targets in both continuous and discontinuous cases. We also extend the VSK framework to multidimensional scaling functions for feature augmentation in kernel regularization networks and SVM classification.

This chapter includes some conclusions and remarks on the content of the book and on the covered topics. At the same time, a discussion of relevant topics that are not covered by the book is also presented.

Standard kernels with scalar shape parameters lack flexibility for capturing anisotropic data structures in high dimensions. We present two-layered kernels built on a matrix transformation used to learn optimal feature directions. For Matérn kernels, we prove convergence rates depend on the effective dimension of this matrix, with faster rates when null space aligns with invariant subspaces of the target function. The 2L-KOGA algorithm combines kernel optimization via stochastic gradient descent with greedy node selection through VKOGA. Numerical experiments on regression benchmarks show 2L-KOGA outperforms standard kernels when significant dimensions can be reduced, with performance correlating to the concentration of singular values.

The Partition of Unity (PU) method addresses computational complexity and stability issues in global kernel approximation by decomposing the domain into overlapping subdomains. Local interpolants are computed independently on each subdomain and combined via weighted sums. Theoretical analysis shows that PU interpolants preserve local approximation orders in the global fit, with error bounds matching those of local schemes. Efficient data structures like integer-based partitioning enable fast point location and neighbor searches. We demonstrate the method's effectiveness for implicit surface reconstruction from 3D point clouds.

This chapter provides a brief introduction to positive definite kernels and the relationships among their various characterizations. It discusses how radial kernels and their associated reproducing kernel Hilbert spaces can offer an effective and flexible tool for addressing the scattered data interpolation problem.

Large-scale SVM training suffers from computational complexity and memory requirements. Local-to-global SVMs (LGSVMs) address this by decomposing the domain into overlapping patches using semi-metrics (Euclidean or cosine), training local SVM classifiers independently on each subdomain, and combining predictions via partition of unity weights. Theoretical analysis shows LGSVMs preserve local classification accuracy while reducing validation time significantly, and the partition of unity framework ensures smooth global decision boundaries. Numerical experiments on high-dimensional datasets demonstrate competitive accuracy with substantial computational speedup compared to global SVMs.

This chapter provides fundamental error bounds for kernel-based interpolation and introduces classical families of radial basis functions. We present stability analysis through conditioning of kernel matrices and derive pointwise error estimates involving the power function and fill distance. Key examples include infinitely smooth kernels (Gaussian, inverse multiquadrics), finitely smooth kernels (Matérn, Wendland functions), and conditionally positive definite functions (thin-plate splines, radial powers). For each family, we discuss native spaces, approximation orders, and stability properties.

Here we collect all the links to the software written by the authors or their coauthors. For each chapter there is a GitHub repository where MATLAB or PYTHON codes are available. Moreover, we include the list of the papers to which the software refer to.

The ability of an interpolant to reproduce polynomials up to a fixed degree, known as polynomial precision, is often a desirable property, as polynomials are easy to manipulate and possess many useful properties. This chapter focuses on the description of conditionally positive definite kernels, which exhibit the polynomial precision property.

Support Vector Machines (SVMs) solve binary classification by finding the maximum-margin separating hyperplane. In the hard margin setting for linearly separable data, the dual formulation reveals that solutions depend only on support vectors, which are training points on the margin. For non-separable data, soft margins introduce slack variables and penalty parameter C, balancing margin width against misclassification tolerance. The kernel trick enables nonlinear decision boundaries through implicit feature mapping. Brief coverage of alternative classifiers includes Naive Bayes, k-Nearest Neighbors, Random Forests, and Neural Networks.