Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems
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Bibliographic record
Abstract
In this paper we study the distribution of $\mathscr C(A)$ and $\log\mathscr C(A)$, where $\mathscr C(A)$ is a condition number for the linear conic system $Ax\leq 0$, $x\neq 0$, with $A\in\Bbb R^{n\times m}$. For Gaussian matrices A we develop both upper and lower bounds on the decay rates of the distribution tails of $\mathscr C(A)$, showing that ${\bf P}\left[\mathscr C(A)\geq t\right]\sim c/t$ for large t, where c is a factor that depends only on the problem dimensions $(m,n)$. Using these bounds, we derive moment estimates for $\mathscr C(A)$ and $\log\mathscr C(A)$ and prove various limit theorems for the cases where m and/or n are large. Combined with condition number based complexity analyses, our results yield tail information on the distribution of running times for interior-point or relaxation methods designed to solve the feasibility problem $Ax\leq 0$, $x\neq 0$.
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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)
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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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