# Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family?

Details: Let $$\mathcal{G}$$ be a family of radial basis functions, e.g. $$\mathcal{G}=\{\exp(-\tfrac1{2\sigma^2}(x-\mu)^2) : \mu\in\mathbb{R}, \sigma\ge 0\}$$ and let $$\mathcal{G}_{m,\alpha}$$ denote the set of all linear combinations of functions from $$\mathcal{G}$$ with at most $$m$$ terms and whose nonzero coefficients are bounded away from $$\alpha$$ (for completeness, $$\alpha=0$$ is allowed): $$\mathcal{G}_{m,\alpha} = \Bigg\{\sum_{k=1}^m a_k g_k : g_k\in\mathcal{G},\, |a_k|\ge\alpha\cdot1(a_k\ne 0)\Bigg\}.$$

Let also $$\mathcal{F}$$ be a (sufficiently regular) family of functions. Let $$f\in\mathcal{F}$$ and $$g^*_{m,\alpha}$$ be such that $$d(f,g^*_{m,\alpha}) = \inf_{g\in\mathcal{G}_{m,\alpha}} d(f,g),$$

where $$d$$ is some metric on functions. (I am including the case $$\alpha=0$$ for this question. The reason for considering $$\alpha > 0$$ is that I am interested in avoiding pathologies with weights that tend to zero in the above projection as $$m$$ increases.)

I am interested in bounds of the form $$d(f,g^*_{m,\alpha})\le C/m^\beta$$ for some constant $$C$$ (which may or may not depend on $$f$$ and $$\alpha$$--I am mostly focused on the dependence on $$m$$). My questions are as follows:

• In the first place, are there known bounds of this form for nontrivial choices of $$(d,\mathcal{F},C)$$? I am being deliberately vague here in order to understand what types of results along these lines are available.
• Ideally, I am interested in the case where $$\mathcal{F}$$ is all smooth densities on $$\mathbb{R}$$, $$\mathcal{G}_{m,\alpha}$$ is restricted to convex combinations of Gaussians, $$\alpha>0$$, and $$d$$ is a standard probability metric such as Hellinger or total variation.

Some additional background: These types of results are quite standard in the approximation theory literature when $$\mathcal{G}$$ is a set of algebraic or trigonometric polynomials (e.g. Bernstein approximation). Of course, it is well-known that $$d(f,g^*_m)\to0$$ when $$\mathcal{G}$$ is the set of Gaussian densities, however, I am not aware of any quantitative convergence estimates for this family. The one paper I am aware of is Li and Barron (2000) who consider the case $$\alpha=0$$.

• I think that the convergence rate must be at least as good as a kernel density estimate. – Steve Huntsman Feb 16 at 1:59
• This is an interesting idea, but note that the KDE is a random object, whereas the question I am asking is a deterministic one. Is there a way to reconcile this? – JohnA Feb 16 at 2:17