Jean-Claude Fort

Theoretical Aspects of the SOM Algorithm

Abstract The SOM algorithm is very astonishing. On the one hand, it is very simple to write down and to simulate, its practical properties are clear and easy to observe. However, on the other hand, its theoretical properties still remain without proof in the general case, despite the tremendous efforts of several authors. In this paper, we briefly review the previous results and provide some conjectures for future work.

Large deviations and rare events in the study of stochastic algorithms

New asymptotics formulas for the mean exit time from an almost stable domain of a discrete-time Markov process are obtained. An original fast simulation method is also proposed. The mathematical background involves the large deviation theorems and approximations by a diffusion process. We are chiefly concerned with the classical Robbins-Monroe algorithm. The validity of the results are tested on examples from the ALOHA system (a satellite type communication algorithm).

Solving a combinatorial problem via self-organizing process: An application of the Kohonen algorithm to the traveling salesman problem

We present an application of the Kohonen algorithm to the traveling salesman problem: Using only this algorithm, without energy function nor any parameter choosen “ad hoc”, we found good suboptimal tours. We give a neural model version of this algorithm, closer to classical neural networks. This is illustrated with various numerical examples.

A stochastic model of retinotopy: A self organizing process

Following Kohonen and using the Hebb principle, we define a self orgaizing stochastic process, which is a simple modelization of the retinotopy, i.e. the establishment of well-ordered connexions between the retina and the cortex.We give some mathematical results about convergence of this process. These results are illustrated by computer simulations.

SOM's mathematics

Since the discovery of the SOMs by T. Kohonen, many results that provide a better description of their behaviour have been found. Most of them are very convincing, but from a mathematical point of view, only a few are actually proved. In this paper, we make a review of some results that are still to be proved and give some framework to formulate various questions.

Time series forecasting: Obtaining long term trends with self-organizing maps

Kohonen self-organisation maps are a well know classification tool, commonly used in a wide variety of problems, but with limited applications in time series forecasting context. In this paper, we propose a forecasting method specifically designed for multi-dimensional long-term trends prediction, with a double application of the Kohonen algorithm. Practical applications of the method are also presented.

About the Kohonen algorithm: strong or weak self-organization?

The question of self-organization for the Kohonen algorithm is investigated. First the notions of organized states, weak and strong self-organizations are precisely defined. Then, combining mathematical and simulation results we prove that the Kohonen algorithm has not the strong self-organization property at least in two well-known cases: the stimuli space is [0, 1](2), the unit set is a line (resp. a grid) with the two nearest (resp. eight nearest) neighbourhood function. Copyright 1996 Elsevier Science Ltd

Asymptotics of optimal quantizers for some scalar distributions

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.

Asymptotic Behavior of a Markovian Stochastic Algorithm with Constant Step

We first derive from abstract results on Feller transition kernels that, under some mild assumptions, a Markov stochastic algorithm with constant step size

Double quantization of the regressor space for long-term time series prediction: method and proof of stability

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