A quadtree, a Steiner spanner, and approximate nearest neighbours in hyperbolic space
Bibliographic record
Abstract
We propose a data structure for point sets in $d$-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces.Using these quadtrees and the L-order we build geometric spanners. Near-linear size $(1+\varepsilon)$-spanners do not exist in hyperbolic spaces, but we create a Steiner spanner that achieves a spanning ratio of $1+\varepsilon$ with $\mathcal O_{d,\varepsilon}(n)$ edges, using a simple construction that can be maintained dynamically. As a corollary, we also get a $(2+\varepsilon)$-spanner (in the classical sense) of the same size, where the spanning ratio $2+\varepsilon$ is almost optimal among spanners of subquadratic size.Finally, we show that our Steiner spanner directly provides a solution to the approximate nearest neighbour problem: given a point set $P$ in $d$-dimensional hyperbolic space we build the data structure in $\mathcal O_{d,\varepsilon}(n\log n)$ time, using $\mathcal O_{d,\varepsilon}(n)$ space. Then for any query point $q$ we can find a point $p\in P$ that is at most $1+\varepsilon$ times farther from $q$ than its nearest neighbour in $P$ in $\mathcal O_{d,\varepsilon}(\log n)$ time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time $\mathcal O_{d,\varepsilon}(\log n)$. This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.
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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 | 0.003 | 0.003 |
| 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 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".