Lorentzian Neural

{We consider neural units whose response functions are Lorentzians rather than the usual sigmoids or steps. This consideration is justiied by the fact that neurons can be paired and that a suitable diierence of the sigmoids of the paired neurons can create a window response function. Lorentzians are special cases of such windows and we take advantage of their simplicity to generate polynomial equations for several problems such as: i)xed points of a completely connected net, ii)classiication of operational modes, iii)training of a feedforward net, iv)process signals represented by complex numbers. 1 1. Introduction The central issue of any theory of neural nets is to nd the values of the synaptic weights which are best suited for a given task. This issue also implies the search of the corresponding best architecture of the net. (To see the close relation between the synaptic weights and the architecture of a net, it suuces to recall that, for example, the suitable cancellation of a suf-cient number of synaptic weights in an initially completely connected net can easily convert the net into a pure feedforward, multilayer net.) Many algorithms (see, e. for the tiling algorithm) have been proposed for the training of a net, including simulated annealing (Kirpatrick, Gelatt & Vecchi, 1983) if local minima of the cost function to be optimized must be avoided. Despite the success of many such algorithms, there is still room for progress to be made in the understanding of the exact nature of operating modes of the nets, such as basins of attractions, (in)stabilities of xed points, elimination of spurious attractors, ne tuning of synaptic weights in order to bring actual responses of the nets closer to their desired targets, etc. The purpose of this paper is to take advantage of the theory of analytical functions to attenuate the diiculties involved in studying these operating modes, as caused by the nonlinearity of the neurons. Thus, we will use neuron units whose responses are meromorphic functions, and actually, for the sake of maximum simplicity, we will only use \Lorentzian" functions. Our aim is to reduce neural net problems to sets of polynomial equations, because of their simpler formal, and often practical, manipulation. EEcient methods (Manocha, 1994) are available to handle such polynomial systems. We will show that, when compared to the manipulation of the usual tran-scendental sigmoid functions, a detailed understanding of such \polynomial" nets can be reached more …