On Combinatorial Depth Measures.
Bibliographic record
Abstract
Given a set P = {p1,..., pn} of points and a point q in the plane, we define a function ψ(q) that provides a combinatorial characterization of the multiset of values {|P ∩Hi|}, where for each i ∈ {1,..., n}, Hi is the open half-plane determined by q and pi. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of ψ(q). The perihedral and eutomic depths of q with respect to P correspond respectively to the num-ber of subsets of P whose convex hull contains q, and the number of combinatorially distinct bisections of P determined by a line through q. We present algorithms to compute the depth of an arbitrary query point in O(n log n) time and medians (deepest points) with re-spect to these depth measures in O(n4) and O(n8/3) time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition in-volves similar combinatorial complexity. 1
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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 | 0.003 | 0.028 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.006 | 0.006 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.006 | 0.017 |
| Open science | 0.003 | 0.006 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".