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 $r_n = o(1)$.

翻译：设 $G_n$ 为基于 $n$ 个独立同分布随机向量 $X_1,\ldots,X_n$（取自 $\R^d$ 上未知密度 $f$）且顶点集为 $[n]$ 的随机几何图。当 $\|X_i -X_j\| \le r_n$ 时存在边 $(i,j)$，其中阈值 $r_n$ 可能依赖于 $n$，$\| \cdot \|$ 表示欧几里得距离。我们研究在仅能访问图的邻接矩阵而未知 $r_n$ 或向量 $X_i$ 时，估计底层空间维度 $d$ 的问题。本文主要结果表明：当 $n^{3/2} r_n^d \to \infty$ 且 $r_n = o(1)$ 时，对于所有满足 $\int f^5 < \infty$ 的密度函数，存在一个估计量 $\hat{d}$ 使得当 $n \to \infty$ 时 $\hat{d}$ 依概率收敛于 $d$。该条件允许非常稀疏的图，因为当 $n^{3/2} r_n^d \to 0$ 时，该图高概率仅包含孤立边。我们还证明，在无需对密度施加任何条件的情况下，当 $n r_n^d \to \infty$ 且 $r_n = o(1)$ 时，存在 $d$ 的一致估计量。