Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces
Summary: We find probability error bounds for approximations of functions \(f\) in a separable reproducing kernel Hilbert space \(\mathcal{H}\) with reproducing kernel \(K\) on a base space \(X\), firstly in terms of finite linear combinations of functions of type \(K_{x_i}\) and then in terms of the projection \(\pi_x^n\) on \(\text{span} \{ K_{x_i}\}_{i=1}^n\), for random sequences of points \(x= (x_i)_i\) in \(X\). Given a probability measure \(P\), letting \(P_K\) be the measure defined by \(\text{d} P_K(x)=K(x,x)\text{d}P(x)\), \(x\in X\), our approach is based on the nonexpansive operator \(L^2(X; P_K)\ni \lambda \mapsto L_{P,K} \lambda:= \int_X\lambda (x) K_x\text{d}P(x)\in\mathcal{H}\), where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by \(\mathcal{H}_P\), that is the operator range of \(L_{P,K}\). Our main result establishes bounds, in terms of the operator \(L_{P,K}\), on the probability that the Hilbert space distance between an arbitrary function \(f\) in \(\mathcal{H}\) and linear combinations of functions of type \(K_{x_i}\), for \((x_i)_i\) sampled independently from \(P\), falls below a given threshold. For sequences of points \((x_i)_{i=1}^\infty\) constituting a so-called uniqueness set, the orthogonal projections \(\pi_x^n\) to \(\text{span} \{K_{x_i}\}_{i=1}^n\) converge in the strong operator topology to the identity operator. We prove that, under the assumption that \(\mathcal{H}_P\) is dense in \(\mathcal{H}\), any sequence of points sampled independently from \(P\) yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or \(L^p\) norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space \(H^2 (\mathbb{D})\) are presented as well.
scientific article; zbMATH DE number 7355395