On the Wasserstein distance between mutually singular measures
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it