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On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line
- Publication Year :
- 2022
-
Abstract
- Let $x_1 < \dots < x_n$ and $y_1 < \dots < y_n$, $n \in \mathbb N$, be real numbers. We show by an example that the assignment problem $$ \max_{\sigma \in S_n} F_\sigma(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^\alpha \, |y_{\sigma(i)} - y_{\sigma(k)}|^\alpha, \quad \alpha >0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n > 2 +2^\alpha$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\text{id}(x,y)$ and $F_\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.
- Subjects :
- Mathematics - Numerical Analysis
Mathematics - Optimization and Control
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2205.09006
- Document Type :
- Working Paper