We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs RGGs with Gaussian d-dimensional latent vectors while only a subset of edges carries latent information determined by a random mask with i.i.d. Bern(q) entries. For any fixed edge density p in (0,1) we determine essentially tight thresholds for this problem as a function of d and q. Our results show that the detection problem is substantially easier if the mask is known upfront compared to the case where the mask is hidden. Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations which arise after approximating characteristic functions by an appropriate power series. The resulting bounds are applicable to much larger subgraphs than considered in previous work which enables tight information-theoretic bounds, while the bounds considered in previous works only lead to lower bounds from the lens of low-degree polynomials. As a consequence we identify the optimal information-theoretic thresholds and rule out computational-statistical gaps. Our bounds further improve upon the bounds on Fourier coefficients of random geometric graphs recently given by Bangachev and Bresler [STOC'24] in the dense, bipartite case. The techniques also extend to sparser and non-bipartite settings, at least if the considered subgraphs are sufficiently small. We furhter believe that they might help resolve open questions for related detection problems.

翻译：本研究探讨了二部随机几何图（RGGs）中隐空间几何结构可检测性的信息论相变问题。图中节点的高斯d维隐向量仅通过随机掩码（其条目为独立同分布的Bern(q)）决定边的子集是否携带隐空间信息。对于任意固定的边密度p∈(0,1)，我们基于维度d和掩码参数q建立了该问题本质上紧致的检测阈值。结果表明：相较于掩码未知的情形，当掩码先验已知时检测问题将显著简化。我们的分析建立在一个新颖的傅里叶分析框架之上，该框架通过将特征函数近似为适当的幂级数，利用高斯随机几何图中带符号子图计数产生的相消效应，推导出新的界。所得边界适用于比以往研究更大的子图规模，从而能建立紧致的信息论界，而先前工作提出的边界仅能基于低阶多项式视角给出下界。由此我们确定了最优信息论阈值，并排除了计算-统计间隙的存在。我们的边界进一步改进了Bangachev与Bresler[STOC'24]近期针对稠密二部情形给出的随机几何图傅里叶系数边界。该技术可扩展至更稀疏与非二部场景，至少当所考虑的子图规模足够小时成立。我们相信这些方法有望为解决相关检测问题中的开放性问题提供帮助。