论文标题
关于与Gromov-Wasserstein距离的分配问题
On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line
论文作者
论文摘要
令$ x_1 <\ dots <x_n $和$ y_1 <\ dots <y_n $，$ n \ in \ mathbb n $，是实数。我们通过一个示例表明，分配问题$ \ max_ {σ\ in s_n}f_σ（x，y）：= \ frac12 \ sum_ {i，k = 1}^n | x_i -x_i -x_k |^|^|^α\，|如果$ n> 2 +2^α$，则由相同的置换（ID）或反性置换（A-ID）求解。实际上，上述最大可以是，取决于点数，任意远离$ f_ \ text {id}（x，y）$和$ f_ \ text {a-id}（x，x，y）$。处理此类任务问题的动机来自它们与Gromov-Wasserstein Diverence的关系，这些分歧最近引起了很多关注。
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 $$ \max_{σ\in S_n} F_σ(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^α\, |y_{σ(i)} - y_{σ(k)}|^α, \quad α>0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n > 2 +2^α$. 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.
