Reduction of symmetric semidefinite programs using the regular*-representation
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Bibliographic record
Abstract
We consider semidefinite programming problems on which a permutation group is acting.We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix *-representation of the commutant (centralizer ring) of thematrix algebra generated by the permutation matrices. We apply it to extending a method of de Klerk et al. that gives a semidefinite programming lowerbound to the crossing number of complete bipartite graphs. It implies that cr( K8,n)> = 2.9299n2-6n,cr ( K9,n)> = 3.8676n2- 8n, and (for any m> = 9) limn!1 cr(Km,n)Z(m, n)> = 0.8594 mm- 1, where Z(m, n) is the Zarankiewicz number b 14 (m- 1)2cb 14 (n- 1)2c, which is the conjectured valueof cr( Km,n). Here the best factor previously known was 0.8303 instead of 0.8594.
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.003 | 0.006 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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