Radial Basis Function (RBF)

AKA: Gaussian Kernel.
Context:
- It can be a Vector Function of the form [math]\displaystyle{ x \longmapsto g(\parallel \mathbf{x} − \mathbf{a} \parallel) }[/math] where [math]\displaystyle{ g }[/math] is a univariate function, [math]\displaystyle{ \mathbf{a} }[/math] is a point in [math]\displaystyle{ \R^d }[/math], and [math]\displaystyle{ \parallel·\parallel }[/math] denotes the Euclidean norm in [math]\displaystyle{ \R^d }[/math].
- It can be used as a Radial Basis Kernel Function, Radial Basis Function Neural Network, ...
- ...
Example(s):
- a Gaussian Radial Basis Function.
- a Multiquadric Radial Basis Function, [math]\displaystyle{ \phi(r) = \sqrt{1 + (\varepsilon r)^2} }[/math]
- a Inverse quadratic Radial Basis Function, [math]\displaystyle{ \phi(r) = \frac{1}{1+(\varepsilon r)^2} }[/math]
- a Inverse Multiquadric Radial Basis Function, [math]\displaystyle{ \phi(r) = \frac{1}{\sqrt{1 + (\varepsilon r)^2}} }[/math]
- a Polyharmonic Radial Basis Function,
  [math]\displaystyle{ \phi(r) = r^k,\; k=1,3,5,\dots }[/math]
  [math]\displaystyle{ \phi(r) = r^k \ln(r),\; k=2,4,6,\dots }[/math]
- ...
Counter-Example(s):
- a Logistic Function.
See: Distance Function, Łukaszyk–Karmowski Metric, Condition Number.

References

http://slideshot.epfl.ch/play/k5FuJcUA0L0c Scaleable Gaussian Processes for Scientific Discovery
- QUOTE: … has fairly strong smoothness assumptions … RBF AKA squared exponential AKA the Gaussian kernel. This is a translation-invariant kernel …
  minute 9:13

(Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/radial_basis_function Retrieved:2014-9-22.
- A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that [math]\displaystyle{ \phi(\mathbf{x}) = \phi(\|\mathbf{x}\|) }[/math]; or alternatively on the distance from some other point c, called a center, so that [math]\displaystyle{ \phi(\mathbf{x}, \mathbf{c}) = \phi(\|\mathbf{x}-\mathbf{c}\|) }[/math]. Any function [math]\displaystyle{ \phi }[/math] that satisfies the property [math]\displaystyle{ \phi(\mathbf{x}) = \phi(\|\mathbf{x}\|) }[/math] is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible. For example, using Łukaszyk–Karmowski metric, it is possible for some radial functions to avoid problems with ill conditioning of the matrix solved to determine coefficients w_i (see below), since the [math]\displaystyle{ \|\mathbf{x}\| }[/math] is always greater than zero. ^[1] Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally invented, by David Broomhead and David Lowe in 1988. ^[2] ^[3] RBFs are also used as a kernel in support vector classification.

(de Boor et al., 2007) ⇒ Carl de Boor, Allan Pinkus, and Vilmos Totik. (2007). “Concepts of Approximation Theory.” In: Surveys in Approximation Theory Web Site, July 21, 2007.
- QUOTE: radial basis function is a function in [math]\displaystyle{ \R^d }[/math] of the form [math]\displaystyle{ x \longmapsto g(\parallel \mathbf{x} − \mathbf{a} \parallel) }[/math] where [math]\displaystyle{ g }[/math] is a univariate function, [math]\displaystyle{ \mathbf{a} }[/math] is a point in [math]\displaystyle{ \R^d }[/math], and [math]\displaystyle{ \parallel·\parallel }[/math] denotes the Euclidean norm in [math]\displaystyle{ \R^d }[/math].