随机几何图在多项式平均度下的尖锐检测阈值
Sharp detection thresholds for random geometric graphs with polynomial average degrees
浏览论文内容
中文总结 AI 辅助
本文证明了随机几何图与 Erdös-Rényi 图在稀疏区域（平均度多项式增长）的不可区分维度阈值猜想，通过后验分析、截断二阶矩及空腔递推方法建立了 KL 散度消失的条件。
中文摘要 AI 辅助
我们研究了由 $\mathbb S^{d-1}$ 上 $n$ 个独立均匀点构成的随机几何图，当它们的内积超过一个为达到边密度 $p$ 而选择的阈值时，两点相连。我们确立了在整个稀疏区域（平均度以多项式速度增长）中，该图与 Erdös-Rényi 图 $G(n,p)$ 不可区分的维度阈值的猜想。具体而言，对于每个固定的 $\delta>0$，若 $p\to0$，$p\ge n^{-1+\delta}$，且 $d\gg (np\log(1/p))^3$，则两种分布之间的 Kullback-Leibler (KL) 散度趋于零。我们的证明基于作者先前工作中引入的后验分析框架，将截断的二阶矩分析与傅里叶展开和球谐展开相结合。与先前工作类似，我们将 KL 界归结为控制一个截断的 $\chi^2$ 散度 $\Delta$，该散度针对最后一个顶点的邻域，条件于由前面顶点诱导的图。然后，我们利用该邻域的条件似然比的傅里叶展开来界定 $\Delta$，其中高频部分通过截断到典型度事件而变得可忽略，而球谐展开则用有界度调和多项式的后验矩来近似低频部分。关键观察是，这些矩和 $\Delta$ 满足一个耦合的空腔递推系统，产生一个自界不等式，从而推出所需的 KL 界。
英文摘要
We study the random geometric graph formed by connecting $n$ independent uniform points on $\mathbb S^{d-1}$ whenever their inner product exceeds a threshold chosen to give edge density $p$. We establish the conjectured dimensionality threshold for indistinguishability from the Erdös-Rényi graph $G(n,p)$ throughout the sparse regime with polynomially growing average degree. Specifically, for every fixed $δ>0$, if $p\to0$, $p\ge n^{-1+δ}$, and $d\gg (np\log(1/p))^3$, then the Kullback-Leibler (KL) divergence between the two distributions vanishes. Our proof builds on the posterior analysis framework introduced in previous work of the authors, combining a truncated second moment analysis with Fourier and spherical harmonic expansions. As in prior works, we reduce the KL bound to controlling a truncated $χ^2$-divergence $Δ$ for the neighborhood of the last vertex conditional on the graph induced by the preceding vertices. We then bound $Δ$ using a Fourier expansion of the conditional likelihood ratio for this neighborhood, where the high-frequency contribution is made negligible by truncation to a typical-degree event, while a spherical harmonic expansion approximates the low-frequency contribution in terms of posterior moments of bounded-degree harmonic polynomials. The key observation is that these moments and $Δ$ satisfy a coupled system of cavity recurrences, yielding a self-bounding inequality that implies the desired KL bound.
发表机构
- Massachusetts Institute of Technology（麻省理工学院）
- Georgia Institute of Technology（佐治亚理工学院）
- Yale University（耶鲁大学）
- Duke University（杜克大学）
机构由 AI 辅助整理，请以论文原文为准。
