On the critical points of the 1-dimensional ocmpetitive learning vector quantization algorithm

摘要

In this contribution is established a specific property of the Competitive Learning Vector Quantization algorithm (also known as the Kohonen algorithm with 0 neighbor): in the 1-dimensional setting, that is when the examples w(sub)t to be coded are scalar with distribution mu, uniquenes of the equilibrium point is established under ln-concavity assumptions on the density f of the distribution mu. The proof relies on the celebrated (finite-dimensional) Mountain pass Lemma. A counter-example is exhibited when f does not satisfy this assumption.