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 $r_n = o(1)$.

给定以未知密度f为基础的n个i.i.d.随机向量Xi产生的随机几何图Gn，估计其潜在空间的维度d的问题。研究发现，在满足条件n^(3/2)r_n^d→∞和r_n=o(1)的情况下，存在一个估计器，其在概率意义下收敛于d，对于所有满足∫f^5<∞的密度函数。条件允许极其稀疏的图，当n^(3/2)r_n^d→0时，图仅包含孤立边的概率很高。同时证明，在不对密度函数做任何条件的情况下，当n r_n^d→∞和r_n=o(1)时，存在一个一致估计器来估计d。

关于从随机几何图中估算维度的注记