实直线上与Gromov-Wasserstein距离有关的指派问题
On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line
摘要：
设$x_1 < \dots < x_n$和$y_1 < \dots < y_n$、$n \in \mathbb N$为实数。我们通过一个例子表明，如果$n > 2 +2^α$满足，则指派问题$$ \max_{σ\in S_n} F_σ(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^α\, |y_{σ(i)} - y_{σ(k)}|^α, \quad α>0, $$一般既不能用恒等置换（id）也不能用反恒等置换（a-id）求解.实际上，根据点的数量，上述最大值可以是任意远离$F_\text{id}(x,y)$和$F_\text{a-id}(x,y)$的。处理这样的分配问题的动机来自于它们与Gromov-Wasserstein分歧的关系，Gromov-Wasserstein分歧最近得到了很多关注。
英文摘要：
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.
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