Probabilistic D-Clustering
giving the minimizer of f(c) =
N
i=1
d(xi, c). Formula (33) can be used
iteratively to update the center c (on the left) as a convex combination of
the points xi with weights depending on the current center. This iteration is
the Weiszfeld method [20] for solving the Fermat–Weber location problem,
see [20], [16]. Convergence of Weiszfeld’s method was established in Kuhn
[15] by modifying the gradient ∇f(c) so that it is always defined, see [17]
for further details. However, the modification is not carried out in practice
since, as shown by Kuhn, the set of initial points c for which it ever becomes
necessary is denumerable.
In what follows we use the formulas (24)–(25) iteratively to update
the centers. Convergence can be proved by adapting the arguments of Kuhn
[15], but as there it requires no special steps in practice.