Hyperplane separability and convexity of probabilistic point sets
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.003 | 0.007 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".