Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability, and dimension selection (Q1108237)

We analyse a Markovian algorithm for the formation of topologically correct features maps proposed earlier by \textit{T. Kohonen} [ibid. 43, 59- 69 (1982; Zbl 0466.92002), and ``Selforganization and associative memory.'' (1984; Zbl 0528.68062)]. The maps from a space of input signals onto an array of formal neurons are generated by a learning scheme driven by a random sequence of input samples. The learning is described by an equivalent Fokker-Planck equation. Convergence to an equilibrium map can be ensured by a criterion for the time dependence of the learning step size. We investigate the stability of the equilibrium map and calculate the fluctuations around it. We also study an instability responsible for a phenomenon termed by Kohonen ``automatic selection of feature dimensions''.