On a Projected Weiszfeld Algorithm

Sebastian Neumayer, Max Nimmer, Gabriele Steidl, Henrike Stephani

Abstract: Weiszfeld’s method provides an efficient way for computing the weighted median of anchor points \(p_j \in {\mathbb {R}}^n\), \(j=1,\ldots ,M\), i.e., the minimizer of \(\sum _{j=1}^M w_j \Vert x-p_j\Vert _2\). In certain applications as in unmixing problems it may be required that x lies in addition in a specific set. In other words we have to deal with a constrained median problem. In this paper we are concerned with closed convex sets lying in a hyperplane of \({\mathbb {R}}^n\) as e.g., the linearly transformed probability simplex. We propose a projected version of the Weiszfeld algorithm to find the minimizer of the constrained median problem and prove the convergence of the algorithm. Here the main contribution is the appropriate handling of iterates taking values in the set of anchor points. A first artificial example shows that the model produces promising results in the presence of different kinds of noise.