Radial Basis Function Network (RBF Network)

Question

In the Wikipedia article on radial basis function network, I didn't understand what was meant by "center vector for neuron i", in other words "center of the RBF units called also prototype".

Answer 1

In RBF networks each neuron in the hidden layer applies a computation that is related to its "center vector".

First consider the set of input neurons as a vector $\mathbf{x}$, and that each hidden layer neuron receives the complete input vector as its own input.

Second, each hidden layer neuron $i$ is parametrised through a vector (center vector) $\mathbf{c_i}$ of equal dimension than $\mathbf{x}$.

The computation of each hidden layer neuron $i$ consists of:

1. Evaluate the distance (according to a metric, which may or may not be Euclidean) between the input $\mathbf{x}$ and the center vector $\mathbf{c_i}$. When the input is equal to the center vector, the output for that neuron will be maximal (see 2).

2. Evaluate a Gaussian function that decays with increasing distance between the vectors. That's the output of each hidden layer neuron. In the following equation, $\beta$ is a parameter that specifies the decay rate of the Gaussian (converse to the standard deviation in a regular Gaussian distribution).

\begin{equation} \rho_i(\mathbf{x})=\exp{[-\beta\Vert\mathbf{x}-\mathbf{c_i}\Vert^2]} \end{equation}

Now combine linearly the output of all hidden layer neurons to obtain the total neural network output. For that, you need an additional set of parameters or weights $a_i$ (one $a_i$ for each hidden layer neuron, but the $a_i$ values are scalars while each $\mathbf{c_i}$ is a vector).

The final equation, that yields the total network output (a scalar value), is:

\begin{equation} \varphi(\mathbf{x}) = \sum_{i=1}^N a_i \rho(||\mathbf{x}-\mathbf{c}_i||) \end{equation}

And that's it.