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Concept
Radial basis function network
Summary
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.
Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers . The output of the network is then a scalar function of the input vector, , and is given by
where is the number of neurons in the hidden layer, is the center vector for neuron , and is the weight of neuron in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian
The Gaussian basis functions are local to the center vector in the sense that
i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron.
Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision.
The parameters , , and are determined in a manner that optimizes the fit between and the data.
In addition to the above unnormalized architecture, RBF networks can be normalized.
Official source
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