Shape recognition via Wasserstein distance
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Bibliographic record
Abstract
The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript d plus 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^{d + 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by computing the cheapest way to transport the mass of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the cost per unit mass transported is a given function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">c\left ( x, y \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript 2 d plus 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^{2d + 2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Motivated by applications to shape recognition, we analyze this transportation problem with the cost <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis x comma y right-parenthesis equals StartAbsoluteValue x minus y EndAbsoluteValue squared"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">c\left ( x, y \right ) = {\left | {x - y} \right |^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and measures supported on two curves in the plane, or more generally on the boundaries of two domains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega comma normal upper Lamda subset-of upper R Superscript d plus 1"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega , \Lambda \subset {R^{d + 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Unlike the theory for measures that are absolutely continuous with respect to Lebesgue, it turns out not to be the case that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu minus a period e period x element-of partial-differential normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mi> μ </mml:mi> <mml:mo> − </mml:mo> <mml:mi>a</mml:mi> <mml:mo>.</mml:mo> <mml:mi>e</mml:mi> <mml:mo>.</mml:mo> <mml:mi>x</mml
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 it