Independence of synthetic Curvature Dimension conditions on transport\n distance exponent
Bibliographic record
Abstract
The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound K joint with an upper bound N on the dimension.Their condition, called the Curvature-Dimension condition and denoted by CD(K, N ), is formulated in terms of a modified displacement convexity of an entropy functional along W 2 -Wasserstein geodesics.We show that the choice of the squared-distance function as transport cost does not influence the theory.By denoting with CD p (K, N ) the analogous condition but with the cost as the p th power of the distance, we show that CD p (K, N ) are all equivalent conditions for any p > 1 -at least in spaces whose geodesics do not branch.Following Cavalletti and Milman [The Globalization Theorem for the Curvature Dimension Condition, preprint, arXiv:1612.07623],we show that the trait d'union between all the seemingly unrelated CD p (K, N ) conditions is the needle decomposition or localization technique associated to the L 1 -optimal transport problem.We also establish the local-to-global property of CD p (K, N ) spaces. Contents1. Introduction 1 2. Prerequisites 5 3. Hopf-Lax transform with exponent p 11 4. Curvature-Dimension conditions: From p > 1 to p = 1 26 5. Curvature-Dimension conditions: From p = 1 to q > 1 3 2 References 45
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".