Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Abstract

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer's classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T k : L 2 (ν) → L 2 (ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer's theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer's theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer's theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L ∞ (ν). 364 Constr Approx (2012) 35:363-417 For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L 2 (ν) → L 2 (ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.

A short note on the comparison of interpolation widths, entropy numbers, and Kolmogorov widths

“…"⇐": By Lemma 4.1, we already know that d n (S : H → L q (µ)) ≺ n −1/α . Moreover, by (19), (10), and (6) we obtain d n (S : H → L 2 (µ)) ≍ n −1/α , and hence we find…”

An optimal kernel‐based U‐statistic method for quantitative gene‐set association analysis

“…Assume is a kernel function defined on . Then the spectral decomposition theorems (Lemma 1 of Chapter 2; Steinwart & Scovel, ) implies that the standardized kernel enjoys the following representation: where the eigenfunctions form a complete orthonormal system (i.e., for any , for ), and are the nonzero eigenvalues satisfying . The standardization is required because could diverge in the high‐dimensional case, and it ensures so that the eigen decomposition can be properly defined.…”

“…Assume ⋅ ⋅ K ( , ) is a kernel function defined on × . Then the spectral decomposition theorems (Lemma 1 of Chapter 2; Steinwart & Scovel, 2012) implies that the standardized kernel ⋅ ⋅ ( , ) enjoys the following representation:…”

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

“…Recall that () is an orthonormal basis (ONB) for , the RKHS associated with the kernel K , where the numbers are the same as in Lemma . Lemma For the embedding we have where the numbers are defined in .…”

“…Since we are dealing with positive definite and continuous kernels on compact sets, we have a Mercer representation for K in terms of the eigenvalues (of the compact integral operator generated by K ) , , and the orthonormal basis (see ) where the convergence is uniform and absolute on . Let be the (compact) integral operator generated by K , that is, …”