On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".