On the interval of fluctuation of the singular values of random matrices
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Bibliographic record
Abstract
Let A be a matrix whose columns X_1,\dots, X_N are independent random vectors in \mathbb R^n . Assume that the tails of the 1-dimensional marginals decay as \mathbb P(|\langle X_i, a\rangle|\geq t)\leq C t^{-p} uniformly in a\in S^{n-1} and i\leq N . Then for p>4 we prove that with high probability A/\sqrt{n} has the Restricted Isometry Property (RIP) provided that Euclidean norms |X_i| are concentrated around \sqrt{n} . We also show that the covariance matrix is well approximated by empirical covariance matrices and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio n/N . Moreover, we obtain sharp bounds for both problems when the decay is of the type exp (-t^{\alpha}) , with \alpha \in (0,2] , extending the known case \alpha \in (1,2] .
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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 | 0.004 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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