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?

J.Galt

Posted 2020-07-26T09:54:38.987

Reputation: 123

1

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

– Nikos M. – 2020-07-26T13:59:38.603

No answers