论文标题
Kantorovich解决方案的内在稀疏性
Intrinsic Sparsity of Kantorovich Solutions
论文作者
论文摘要
令$ x，y $为两组有限的积分，其中$ \＃x = m $和$ \＃y = n $点，$μ=（1/m）\ sum_ {i = 1}^{m}^{m}δ_{x_i} $措施。 Birkhoff的结果意味着，如果$ M = N $，那么Kantorovich问题也具有解决蒙格问题的解决方案：可以使用Bookjection $π来实现最佳运输：X \ Rightarrow y $。当$ m \ neq n $时，这是不可能的。我们观察到，当$ m \ neq n $时，就会有一个解决方案kantorovich问题的解决方案，使得$ x $中的每个点的质量最多移至$ n/\ gcd（m，n）$ y $中的$ y $中的不同点，并且对此，对话，每个点在$ y $中的每一个$ y $ in $ y $最多从$ m/\ gcd（m/\ gcd n）$ x $ x $ x $ x in $ x in $ x。
Let $X,Y$ be two finite sets of points having $\#X = m$ and $\#Y = n$ points with $μ= (1/m) \sum_{i=1}^{m} δ_{x_i}$ and $ν= (1/n) \sum_{j=1}^{n} δ_{y_j}$ being the associated uniform probability measures. A result of Birkhoff implies that if $m = n$, then the Kantorovich problem has a solution which also solves the Monge problem: optimal transport can be realized with a bijection $π: X \rightarrow Y$. This is impossible when $m \neq n$. We observe that when $m \neq n$, there exists a solution of the Kantorovich problem such that the mass of each point in $X$ is moved to at most $n/\gcd(m,n)$ different points in $Y$ and that, conversely, each point in $Y$ receives mass from at most $m/\gcd(m,n)$ points in $X$.
