Multivariate sharp quadratic bounds via $\mathbf{\Sigma}$-strong convexity and the Fenchel connection
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Bibliographic record
Abstract
Sharp majorization is extended to the multivariate case. To achieve this, the notions of $\sigma$-strong convexity, monotonicity, and one-sided Lipschitz continuity are extended to $\mathbf{\Sigma}$-strong convexity, monotonicity, and Lipschitz continuity, respectively. The connection between a convex function and its Fenchel-Legendre transform is then developed. Sharp majorization is illustrated in single and multiple dimensions, and we show that these extensions yield improvements on bounds given within the literature. The new methodology introduced herein is used to develop a variational approximation for the Bayesian multinomial regression model.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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