A class of multi-marginal c-cyclically monotone sets with explicit c-splitting potentials
Bibliographic record
Abstract
Multi-marginal optimal transport plans are concentrated on c-splitting sets. It is known that, similar to the two-marginal case, c-splitting sets are c-cyclically monotone. Within a suitable framework, the converse implication was very recently established by Griessler. However, for an arbitrary cost c, given a multi-marginal c-cyclically monotone set, the question whether there exists an analogous explicit construction to the one from the two-marginal case of c-splitting potentials is still open. When the margins are one-dimensional and the cost belongs to a certain class, Carlier proved that the two-marginal projections of a c-splitting set are monotone. For arbitrary products of sets equipped with cost functions which are sums of two-marginal costs, we show that the two-marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of c-splitting potentials. Our condition is, in principle, easier to verify than the one of multi-marginal c-cyclic monotonicity. Various examples illustrate our results. We show that, in general, our condition is sufficient; however, it is not necessary. On the other hand, we conclude that when the margins are one-dimensional equipped with classical cost functions, our condition is a characterization of c-splitting sets and extends classical convex analysis.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
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".