A note on estimating the dimension from a random geometric graph
Bibliographic record
Abstract
Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold $r_n$ possibly depending upon $n$, where $\| \cdot \|$ denotes Euclidean distance. We study the problem of estimating the dimension $d$ of the underlying space when we have access to the adjacency matrix of the graph but do not know $r_n$ or the vectors $X_i$. The main result of the paper is that there exists an estimator of $d$ that converges to $d$ in probability as $n \to \infty$ for all densities with $\int f^5 < \infty$ whenever $n^{3/2} r_n^d \to \infty$ and $r_n = o(1)$. The conditions allow very sparse graphs since when $n^{3/2} r_n^d \to 0$, the graph contains isolated edges only, with high probability. We also show that, without any condition on the density, a consistent estimator of $d$ exists when $n r_n^d \to \infty$ and $r_n = o(1)$.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".