On the Wasserstein distance between mutually singular measures
Bibliographic record
Abstract
Abstract We study the Wasserstein distance between two measures <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:math> {\mu,\nu} which are mutually singular. In particular, we are interested in minimization problems of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">𝒜</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo movablelimits="false">inf</m:mo> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>ν</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mi>ν</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">𝒜</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> W(\mu,\mathcal{A})=\inf\{W(\mu,\nu):\nu\in\mathcal{A}\}, where μ is a given probability and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} is contained in the class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>μ</m:mi> <m:mo>⊥</m:mo> </m:msup> </m:math> {\mu^{\perp}} of probabilities that are singular with respect to μ. Several cases for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} are considered; in particular, when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} consists of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {L^{1}} densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>min</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>A</m:mi>
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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 | 0.007 | 0.030 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".