# Using an Opportunity Matrix to Select Centers for RBF Neural Networks
## 1 Introduction
### Overview of RBF Neural Networks  
Given the challenges that often arise when training MLP-based neural networks, AI/ML practitioners seeking a solution for high-precision function approximation may benefit greatly from considering another type of neural network architecture--that of a radial basis function (RBF) network [5]. In contrast with conventional MLP networks, RBF neural networks differ in that they rely on a single hidden layer of nodes in which nonlinear radial basis functions are used as the nodes' activation functions. This particulartype of neural network architecture has shown itself to be of value in a wide variety of practical applications, including solving partial differential equations [6], real-time facial recognition [7], computer vision [8], and signal processing [9], among many others.  
Although RBF neural networks most commonly rely on Gaussian radial basis functions, other types of radial basis functions may be substituted when appropriate [10]. The standard Gaussian radial basis function is defined in Equation (1) below [11]. The parameter \(r\) in Equation (1) denotes the distance between an input value and the RBF node's \"center\" (which will be discussed shortly), while the parameter \(\sigma\) controls the shape of the Gaussian. Euclidean distance is ordinarily used for \(r\); however, the Mahalanobis distance measure may yield superior results in certain problem domains, such as pattern recognition [12].  
\[\varphi(r)=e^{-r^{2}/2\sigma^{2}} \tag{1}\]  
After each input value passes through the RBF layer, the weighted outputs from the RBF nodes are linearly combined to yield the RBF network's overall output [13]. An illustrative example of the standard RBF neural network architecture is provided in Figure 1. Importantly, it is only the weights between the RBF layer and the output layer (denoted \(w_{1},\ w_{2},\ \dots\ w_{n}\) in Figure 1) that need to be learned during the network training process.  
Each RBF node in an RBF neural network is characterized by a \"center\" and a \"width\". Each RBF node center takes the form of a vector of the same dimensionality as the input vector and can be conveniently conceptualized as a geometric location within the input space. For a Gaussian RBF node, the width is parameterized as the standard deviation, the value of which controls the rate at which the node's influence diminishes as input values become more and more distant from the node's center. During training, only those few RBF nodes whose centers are closest to an input value will experience any sort of significant activation, the effect of which is that only a small number of weights will be updated in response to the input value. Once the locations of the RBF centers have been chosen, finding optimal weights for an RBF network can be accomplished either through direct calculation by using a pseudoinverse approach [5; 14] or via gradient descent [10; 15]. Notably, the training process for RBF neural networks is often several orders of magnitude faster than for MLP-based neural networks with corresponding degrees of function approximation capability [5; 16; 17].