Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

Abstract

The theories for radial basis functions (RBFs) as well as piecewise polynomial splines have reached a stage of relative maturity as is demonstrated by the recent publication of a number of monographs in either field. However, there remain a number of issues that deserve to be investigated further. For instance, it is well known that both splines and radial basis functions yield "optimal" interpolants, which in the case of radial basis functions are discussed within the so-called native space setting. It is also known that the theory of reproducing kernels provides a common framework for the interpretation of both RBFs and splines. However, the associated reproducing kernel Hilbert spaces (or native spaces) are often not that well understood -especially in the case of radial basis functions. By linking (conditionally) positive definite kernels to Green's functions of differential operators we obtain new insights that enable us to better understand the nature of the native space as a generalized Sobolev space. An additional feature that appears when viewing things from this perspective is the notion of scale built into the definition of these function spaces. Furthermore, the eigenfunction expansion of a positive definite kernel via Mercer's theorem provides a tool for making progress on such important questions as stable computation with flat radial basis functions and dimension independent error bounds.

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

“…Fixing the value at x j seems to prevent values on opposite sides of it from communicating with each other. Given that the Matérn kernel is the Green's function of a PDE (Whittle, 1954(Whittle, , 1963Fasshauer, 2012) this should not come as a surprise, since conditioning on Y essentially creates two independent boundary value problems. Simple linear algebra shows that this effect can always be expected if the precision matrix Θ −1 is (approximately) banded.…”

Uncovering Local Trends in Genetic Effects of Multiple Phenotypes via Functional Linear Models

“…There are relatively few published recommendations in the statistical literature on how to construct . For example, Lian [] writes “[...]the construction of k 1 in general is difficult and a search of the literature does not seem to provide us with any clues about how to construct a positive definite kernel in general.” Nonetheless, if we shift our attention to the machine learning literature, we see that , where is a Green's function of the linear differential operator [Fasshauer, ; Fasshauer and Ye, ; Poggio and Girosi, ; Rasmussen and Williams, ]. Note that the Green's function also depends on the boundary conditions.…”

“…is a Green's function of the linear differential operator L y(t) [Fasshauer, 2012;Fasshauer and Ye, 2013;Poggio and Girosi, 1990;Rasmussen and Williams, 2006]. Note that the Green's function also depends on the boundary conditions.…”

Decomposition and reconstruction of multidimensional signals by radial functions with tension parameters

“…Connections between either splines and Green's functions or radial basis functions and Green's functions have repeatedly been used and made over the past decades (see e.g. [17], [19], [10], and the references therein). A very special case are Polyharmonic splines introduced in [9] that have been well studied by several authors because of their interesting properties (see e.g.…”

“…That is, strictly positive definite functions or conditionally positive definite functions of order 2n − 1 < m that are fundamental solutions of known differential operators. Connecting (conditionally) positive definite kernels to fundamental functions, provides an interpretation of native spaces as generalized Sobolev spaces associated to L (see [10]). Here, we mainly consider the case when the parameters κ ℓ are all positive.…”