Radial basis function: Difference between revisions

Revision as of 12:37, 6 May 2025

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== Definition ==

A radial function is a function <math display="inline">\varphi:[0,\infty) \to \mathbb{R}</math>. When paired with a [[norm on a vector space(mathematics)|norm]] <math display="inline"> \|\cdot\|:V \to [0,\infty)</math> on a vector space, a function of the form <math display="inline"> \varphi_\mathbf{c} = \varphi(\|\mathbf{x}-\mathbf{c}\|) </math> is said to be a '''radial kernel''' centered at <math display="inline"> \mathbf{c} \in V </math>. A radial function and the associated radial kernels are said to be radial basis functions if, for any [[finite set]] of nodes <math>\{\mathbf{x}_k\}_{k=1}^n \subseteq V</math>, all of the following conditions are true:

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| The kernels <math>\varphi_{\mathbf{x}_1}, \varphi_{\mathbf{x}_2}, \dots, \varphi_{\mathbf{x}_n}</math> are linearly independent (for example <math>\varphi(r)=r^2</math> in <math>V=\mathbb{R}</math> is not a radial basis function)