Proc. of Intelligent System and Control (ISC99), Santa-Barbar, October 1999 ABSTRACT We propose a new method called C-SOM using a Self-Organizing Map (SOM) for function approximation. C-SOM takes care about the output values of the «win- ning» neuron's neighbors of the map to compute the output value associated with the input data. Our work extends the standard SOM with a combination of Local Linear Map- ping (LLM) and cubic spline based interpolation techni- ques to improve its generalization capabilities. We use the gradient information provided by the LLM technique to interpolate in the input space between neighboring neurons of the map in order to get a first-order continuity at the bor- der hyperplanes of Voronoï regions between these neurons. We present the case of a one-dimensional map and show this method performs better than standard SOM and stan- dard LLM in different function approximation tests. KEY WORDS Neural Networks; Self-Organizing Map; function approximation; cubic spline. 1. INTRODUCTION Multi Layered Perceptron (MLP) or Radial Basis Function (RBF) neural networks are known for their abili- ties to approximate theoretically any kind of function with the desired accuracy [1]. However, the learning process of that kind of neural networks can get stuck into local minima of its global cost function. This is hardly possible with the local architecture of SOMs using LLM [2] where each neuron has to minimize a local cost function which is quadratic and which has obviously only one minimum. Moreover, in MLPs and RBFs, the minimization of the global cost during the learning phase could sometime increase the local error of the network. That brings about a bothersome oblivion of what it has been learnt before. That kind of drawback is particularly undesirable in on-line reinforcement learning systems [3][4] where no model of the controlled process is available and where it is hardly possible to play the same set of data again and again until the learning convergence. A local learning method such as LLM with a SOM architecture [2], should limit that draw- backs as discussed in [5]. The standard self-organizing maps referred to SOMs [6] have very interesting properties as they realize a projec- tion from a high-dimensional continuous input space onto the low-dimensional discrete space of the map, preserving topology of the input space and density distribution of the data. Moreover, an output vector (or output weight) can be associated to each neuron in a way to realize an associative memory. However, that vector quantization brings about the poor performances of SOMs in function approximation and system identification, because of its discrete represen- tation of the data. SOMs are competitive-like networks which share the input space into clusters called «Voronoï» regions corres- ponding to neurons of the map. Each of these neurons is located in the input space thanks to a kernel weight. Any point of a cluster is projected onto the «closest» correspon- ding neuron of the map according to the Euclidean dis- tance. That kind of projection can be useful in classification process but drags a drawback in function approximation. Indeed, a standard SOM makes no difference between points of a cluster «close» or «far» from the cluster's ker- nel, and assigns to the both the same output vector. Thus, the output «landscape» provided by standard SOM looks like a basaltic (hyper-)pillars field, with a lot of discontinu- ities along the edge of Voronoï regions. It leads us to think that SOMs should interpolate between neighboring regions to get a continuous variation of output values like on a smooth landscape. In that way, SOMs should perform a more accurate function approximation (Figure 1). Figure 1: SOM, LLM and C-SOM's function approxima- tion methods. Examples of discontinuities are encircled. C-SOM: A CONTINUOUS SELF-ORGANIZING MAP FOR FUNCTION APPROXIMATION MICHAEL AUPETIT PIERRE MASSOTTE PIERRE COUTURIER LGI2P - EMA - Site EERIE Parc Scientifique Georges Besse F-30000 Nîmes, France email: {aupetit,massotte,pcouturi}@site-eerie.ema.fr N 1 N 2 N 3 Output space Input space Standard SOM Standard LLM C-SOM Function to approximate Voronoï edge