*Europhys. Lett.*,

**38**(1), pp. 73-78 (1997)

## Dynamics of on-line competitive learning

Institut für Theoretische Physik, Universität Würzburg Am Hubland, D-97074 Würzburg, Germany

Received:
4
November
1996

Accepted:
24
February
1997

We present a solvable model of unsupervised competitive learning, which determines prototype vectors suitable for the representation of high-dimensional data. In the thermodynamic limit, the dynamics of on-line training is described exactly by a system of coupled first-order differential equations for a set of order parameters. As an example application of the formalism we discuss the identification of two prototypes in the case of two overlapping clusters of data. This specific model exhibits non-trivial behavior like almost stationary plateau configurations which correspond to weakly repulsive fixed points of the dynamics. The ability of the system to escape from this fixed point as well as its asymptotic behavior depend critically on the learning rate used in the algorithm.

PACS: 87.10.+e – General, theoretical, and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics, and bionics) / 02.50.-r – Probability theory, stochastic processes, and statistics / 07.05.Kf – Data analysis: algorithms and implementation; data management

*© EDP Sciences, 1997*