## Kernel approximation of a function known only point-wise?

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Assume that I have a set of $$N$$ points $$x_i, i=1,...,N,$$ in some space $$\mathbb{R}^D$$, and corresponding point-wise (scalar) function evaluations $$f(x_i)$$. It is my goal to approximate the unknown function $$f(x)$$ with RBF kernels:

$$\tilde{f}(x)=\sum_{i=1}^N w_i k(x,x_i)$$

where $$k(x,x_i)$$ is a RBF kernel centred on $$x_i$$. It may seem intuitive to set $$w_i=f(x_i)$$, but then I will usually not reclaim $$f(x_i)$$ due to the influence of other basis functions. Towards this end, I have a number of questions, and would appreciate it if you could answer some of them:

1. Is there a procedure you can recommend for finding the weights $$w_i$$ so that $$\tilde{f}(x_i)=f(x_i)$$?
2. Can this procedure work with higher-dimensional output (i.e., $$f:\mathbb{R}^D \rightarrow \mathbb{R}^E,E>1$$)?
3. Are there any references or lectures you can recommend on the topic?

1

have you seen the general method and if so, where are you stuck exactly?

– Nikos M. – 2020-07-26T13:59:38.603