On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals
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arXiv
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| Format: | Preprint |
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2024
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| author | Emami, Pedram Pass, Brendan |
| author_facet | Emami, Pedram Pass, Brendan |
| contents | A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is $m$-empirical (that is, uniformly supported on $m$ points). By direct computation, we provide an example showing that the ansatz \emph{can fail} when the underlying dimension $d$ is $2$, number of marginals $N$ to be matched is $3$ and the size $m$ of their supports is $3$. As a consequence, the set of $m$-empirical measures is not barycentrically convex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well known consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for $N=2$, standard techniques show it holds when $d=1$, and we provide a simple proof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$ in our counterexample are as small as possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals Emami, Pedram Pass, Brendan Optimization and Control A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is $m$-empirical (that is, uniformly supported on $m$ points). By direct computation, we provide an example showing that the ansatz \emph{can fail} when the underlying dimension $d$ is $2$, number of marginals $N$ to be matched is $3$ and the size $m$ of their supports is $3$. As a consequence, the set of $m$-empirical measures is not barycentrically convex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well known consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for $N=2$, standard techniques show it holds when $d=1$, and we provide a simple proof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$ in our counterexample are as small as possible. |
| title | On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2401.12417 |