Hybrid Learning of RBF Networks Roman Neruda ⋆ and Petra Kudov´ a Institute of Computer Science, Academy of Sciences of the Czech Republic, P.O. Box 5, 18207 Prague, Czech Republic roman@cs.cas.cz Abstract. Three different learning methods for RBF networks and their combi- nations are presented. Standard gradient learning, three-step algoritm with un- supervised part, and evolutionary algorithm are introduced. Their perfromance is compared on two benchmark problems: Two spirals and Iris plants. The re- sults show that three-step learning is usually the fastest, while gradient learning achieves better precission. The combination of these two approaches gives best results. 1 Introduction By an RBF unit we mean a neuron with multiple real inputs x =(x 1 ,...,x n ) and one output y. Each unit is determined by an n-dimensional vector  c which is called center. It can have an additional parameter b usually called width. The output y is computed as: y = ϕ(ξ ); ξ = ‖ x −  c ‖ b (1) where ϕ : → is a suitable activation function, typically Gaussian ϕ(z )= e -z 2 . For evaluating ||x- c|| d , the Euclidean norm is usually used. In this paper we consider a general weighted norm instead of the Euclidean norm. A weighted norm is determined by a n × n matrix C and is defined as ‖ x ‖ 2 C =(Cx) T (Cx)= x T C T Cx. (2) It can be seen that the Euclidean norm is a special case of a weighted norm de- termined by an identity matrix. In further text we will use the symbol Σ -1 instead of C T C. In order to use a weighted norm each RBF unit has another additional parameter matrix C. An RBF network is a standard 3-layer feedforward network with the first layer con- sisting of n input units, a hidden layer consisting of h RBF units and an output layer of m linear units. Thus, the network computes the following function  f : n → m : f s (x)= h  j=1 w js ϕ  ‖ x −  c j ‖ Cj b j  , (3) ⋆ This work has been partially supported by GACR under grants 201/00/1489 and 201/02/0428. P.M.A. Sloot et al. (Eds.): ICCS 2002, LNCS 2331, pp. 594-603, 2002.  Springer-Verlag Berlin Heidelberg 2002