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 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 | 0.000 | 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)
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 teacher head, 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".