Function Approximation with Continuous Self - Organizing Maps using Neighboring Influence Interpolation

A new multi-dimensional interpolation method for function approximation using Self-Organizing Maps (SOMs) [1] is proposed. The output is continuous and infinitely differentiable in all the interpolation area, and the error function has no local minima insuring the convergence of the learning toward the minimal error. The complexity is O(n) in the general case (where n is the number of neurons). The novelty of this method lies in two facts. First, it uses directly the Euclidean distances already computed to self-organize the map in the input space. Second, it uses the neighborhood relationships between the neurons to create " neighboring " influence kernels. The interpolation leans on the Local Linear Mapping (LLM) [2] computed by each neuron for which the " neighboring " influence kernel is activated by the input vector. The results obtained in different approximation tests show that this new interpolation method improve the approximation quality of standard SOMs using LLM. A comparison between such a Continuous SOM (CSOM) and a Multi-Layered Perceptron (MLP) is also presented showing it has equivalent performances in term of accuracy and it is liable to limit the interference phenomenon [3] thanks to its local representation. This method could be applied fruitfully to other local networks such as Neural Gas (NG) [4][5] or Growing Neu-ral Gas (GNG) [6][7] networks.