Hyperplane separability and convexity of probabilistic point sets
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Bibliographic record
Abstract
We describe an $O(n^d)$-time algorithm for computing the exact probability that two $d$-dimensional probabilistic point sets are linearly separable, for any fixed $d \geq 2$. A probabilistic point in $d$-space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the $d$-dimensional separability problem is equivalent to a $(d+1)$-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of $n$ probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of $n$. In addition, our algorithms can handle input degeneracies in which more than $k+1$ points may lie on a $k$-dimensional subspace, thus resolving an open problem in Agarwal et al 2013. Finally, we prove lower bounds for the separability problem via a reduction from the $k$-SUM problem, which show in particular that our $O(n^2)$ algorithms for $2$-dimensional separability and $3$-dimensional convex hull membership are nearly optimal.
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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 | 0.001 |
| 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