2022/05/18 by Robert Beinert, Beinert, Robert, Cosmas Heiß +3 · 3 citations
Mathematics · #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2205.09006
Let x1 < … < xn and y1 < … < yn, n ∈ \mathbb N, be real numbers. We show by an example that the assignment problem maxσ∈ Sn Fσ(x,y) := \frac12 ∑i,k=1n |xi - xk|α |yσ(i) - yσ(k)|α, αgt;0, is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if n > 2 +2α. Indeed the above maximum can be, depending on the number of points, arbitrary far away from Fid(x,y) and Fa-id(x,y). The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.