The diameter of the Birkhoff polytope
Bibliographic record
Abstract
Abstract The geometry of the compact convex set of all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>n</m:mi><m:mo>×</m:mo><m:mi>n</m:mi></m:math> n\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup><m:mrow><m:mi>ℓ</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msubsup></m:math> {\ell }_{n}^{p} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup><m:mrow><m:mi>ℓ</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msubsup></m:math> {\ell }_{n}^{p} and the Schatten <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>p</m:mi></m:math> p -norms, both for the range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mn>1</m:mn><m:mo>≤</m:mo><m:mi>p</m:mi><m:mo>≤</m:mo><m:mi>∞</m:mi></m:math> 1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 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".