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Radial Function

Wikipedia: Radial function:

In mathematics, a radial function is a function defined on a Euclidean space $\mathbf{R}^n$ whose value at each point depends only on the distance between that point and the origin.

“radial” 的意思应该是:”只与 radius (半径长度) 有关“。所以有:$\phi (\mathbf{x} )=\phi (\Vert \mathbf{x} \Vert)$

Radial function 的一个特性是:翻转、旋转这类不改变 $\mathbf{x}$ 向量长度的线性变换不会改变 $\phi (\mathbf{x})$ 的值。

Wikipedia: Radial basis function:

… or alternatively on the distance from some other point $c$, called a center, so that $\phi (\mathbf{x} ,\mathbf{c})=\phi (\Vert \mathbf{x} - \mathbf{c} \Vert)$.

这也就是常见的二元 radial function 的形式:$\phi (\mathbf{x} ,\mathbf{y}) = \varphi(r)$ where $r= \Vert \mathbf{x} - \mathbf{y} \Vert$。

这里 $\Vert \mathbf{x} - \mathbf{y} \Vert$ 可以看做从 $\mathbf{y}$ 指向 $\mathbf{x}$ 的一个 radius。

Radial Basis Function

Radial basis function 是 radial function 的子类。所谓 Radial basis function 就是它的定义中会涉及到一个 power (幂),然后 radius ($\Vert \mathbf{x} \Vert$ or $\Vert \mathbf{x} - \mathbf{y} \Vert$) 会是这个幂的 base (基数)。比如:

  • Gaussian RBF: ${\phi (\mathbf{x} ,\mathbf{y}) = \varphi(r) = e^{-(\varepsilon r)^{2}}}$
  • Multiquadric RBF: $\phi (\mathbf{x} ,\mathbf{y}) = \varphi(r) = {\sqrt {1+(\varepsilon r)^{2}}}$

Recap: Base / Exponent / Power

对 $a^n$:

  • $a$ is the base
  • $n$ is the exponent
  • $a^n$ is the power, or precisely the $n^{\text{th}}$ power of $a$

我觉得我对 “power” 这个词的不理解,一是源自 “$a$ raised to the $n^{\text{th}}$ power”、”$a$ raised to the power of $n$” 这类的表达方式。其实记住 power 就是 $a^n$ 这个值就可以了。

二是觉得这个概念用 “power” 这个单词表示很奇怪。其实这里涉及到 etymology 的问题。

Etymology of some common mathematical terms:

Power is first used for the square. Euclid uses the phrase in power, for example he says that magnitudes are commensurable in power when their squares are commensurable.

然后 Etymology of “power” (math.) 提到原词是希腊文 δυνάμει (dunamis),表示 potentiality,与 actuality 相对。所以我觉得 in power 就是 in potentiality 亦即 potentially

“magnitudes are commensurable in power when their squares are commensurable” 这句话里的 magnitudes 用 “incommensurable” 修饰一下就更好理解了。举个例子:

  • $\sqrt 2$ is actually incommensurable.
  • $\sqrt 2$’s square, $2$, is commensurable.
  • So we say $\sqrt 2$ is protentially commensurable.

Etymology of “power” (math.):

Thus, from the Greek dunamis to the Latin potentia and finally to power.

至于 exponent,它并不出自 potentia,而是源自 expōnō,本意是 expose,我觉得它就是描述 $n$ stands out in the notation of $a^n$ 的意思。