In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.

We present a new way of study of Mercer kernels, by corresponding to a special kernel $K$ a pseudo-differential operator $p({\mathbf x}, D)$ such that $\mathcal{F} p({\mathbf x}, D)^\dag p({\mathbf x}, D) \mathcal{F}^{-1}$ acts on smooth functions in the same way as an integral operator associated with $K$ (where $\mathcal{F}$ is the Fourier transform). We show that kernels defined by pseudo-differential operators are able to approximate uniformly any continuous Mercer kernel on a compact set. The symbol $p({\mathbf x}, {\mathbf y})$ encapsulates a lot of useful information about the structure of the Maximum Mean Discrepancy distance defined by the kernel $K$. We approximate $p({\mathbf x}, {\mathbf y})$ with the sum of the first $r$ terms of the Singular Value Decomposition of $p$, denoted by $p_r({\mathbf x}, {\mathbf y})$. If ordered singular values of the integral operator associated with $p({\mathbf x}, {\mathbf y})$ die down rapidly, the MMD distance defined by the new symbol $p_r$ differs from the initial one only slightly. Moreover, the new MMD distance can be interpreted as an aggregated result of comparing $r$ local moments of two probability distributions. The latter results holds under the condition that right singular vectors of the integral operator associated with $p$ are uniformly bounded. But even if this is not satisfied we can still hold that the Hilbert-Schmidt distance between $p$ and $p_r$ vanishes. Thus, we report an interesting phenomenon: the MMD distance measures the difference of two probability distributions with respect to a certain number of local moments, $r^\ast$, and this number $r^\ast$ depends on the speed with which singular values of $p$ die down.

This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of $\mathbb{R}^d$. On this approximate eigenbasis the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as $\mathcal{O}(nm^2)$ (initial) and $\mathcal{O}(m^3)$ (hyperparameter learning) with $m$ basis functions and $n$ data points. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.