Kohonen's algorithm for the numerical parametrisation of manifolds

T. Kohonen has described an algorithm for fitting a k-dimensional grid of points to a set of points taken from a k-manifold in R n , for k⩽ n. The algorithm is inspired by a neural model and bears some of the marks of its ancestry. In this paper we show that if the process converges, it converges to a locally 1-1 mapping of the grid onto the manifold. Hitherto this result has only been proved for the case where k=1.