Radial Function / Radial Basis Function / Base / Exponent / Power

Yao Yao on May 9, 2018

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。

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

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

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

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.

“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.

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