Radial Basis Functions (RBFs)

Radial Basis Functions (RBFs) are a family of functions whose values depend solely on the distance between input points. This distance is often measured from a central point, known as the center (denoted $\mathbf{c}$). When dealing with vectors, the notation $\Vert \mathbf{x}\Vert$ represents the norm or magnitude of the vector $\mathbf{x}$, which in Euclidean space is the square root of the sum of squares of its components. Mathematically, for $\mathbf{x}=(x_1,x_2,\dots ,x_n)$, the Euclidean norm is $\Vert \mathbf{x}\Vert =\sqrt{x_1^2+x_2^2+\cdots +x_n^2}$​​. This is what $\Vert \cdot \Vert$ symbolizes: the “size” or “length” of the vector, which gives RBFs their radial, distance-based nature.

Explanation of RBFs

An RBF takes the form $\phi (\Vert \mathbf{x}-\mathbf{c}\Vert )$, where:

• $\mathbf{x}$ is an input point in space.
• $\mathbf{c}$ is a center point for the RBF.
• $\Vert \mathbf{x}-\mathbf{c}\Vert$ is the distance between $\mathbf{x}$ and $\mathbf{c}$, often Euclidean.

The RBF function itself might be a Gaussian function, like $\phi (r)=e^{-(\epsilon r{)}^2}$, where $r$ represents distance. This RBF will have its highest value when $\mathbf{x}$ is at $\mathbf{c}$ (i.e., when $\Vert \mathbf{x}-\mathbf{c}\Vert =0$), and it diminishes as $\mathbf{x}$ moves away from $\mathbf{c}$, resembling a bell-shaped curve.