Convergence in distributions of the multi-dimensional Kohonen algorithm (Q2731157)

\textit{T. Kohonen} [Biol. Cybern. 43, 59-69 (1982; Zbl 0466.92002)] introduced the self-organizing map (SOM) as a simple model of a process of self-organization between different areas of the cortex and different sensory inputs. See also \textit{T. Kohonen} [``Self-organizing maps'' (1994; Zbl 0827.68092)] and the author [``A mathematical study of self-organizing neural networks'' (Shaker, Aachen)]. The simplicity of its learning algorithm immediately made the map interesting for many artificial intelligence applications, e.g., clustering, feature mapping and building topology preserving maps. Besides the feedforward neural networks, which are designed for supervised learning, SOM is the most well known neural network with a reasonable capability for unsupervised data analysis.NEWLINENEWLINENEWLINEIn the present paper, the author considers the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that this algorithm is neither weak nor strong Feller, he shows that it is a \(T\)-chain, regardless of the dimensionalities of both data space and the special shape of the neighborhood function. Furthermore, in the case of a multidimensional setting, which is practically important, the author shows that the chain is \(\Psi\)-irreducible and aperiodic. He then shows that Doeblin's condition is valid, which ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Finally, he shows that the process is Harris recurrent, which enables the use of statistical devices to measure the centrality and variability of the invariant probability measure.