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：我们研究具有高维高斯潜向量的二分随机几何图的潜结构可检测性中的信息论相变问题，其中仅部分边携带潜信息，这些边由独立同分布伯努利(q)随机掩码决定。对于任意固定边密度p∈(0,1)，我们确定了该问题随d和q变化的基本紧致阈值。结果表明，当掩码已知时，检测问题显著比掩码隐藏时更容易。我们的分析基于一种新颖的傅里叶分析框架，用于界定高斯随机几何图中的符号子图计数，该框架利用了通过适当幂级数逼近特征函数后产生的相消效应。所得边界适用于比先前工作更大的子图结构，从而能获得紧致的信息论边界，而以往研究仅通过低次多项式视角给出下界。由此我们识别出最优信息论阈值，并排除了计算-统计差距。我们的边界进一步改进了Bangachev与Bresler[STOC'24]最近在稠密二分情形下给出的随机几何图傅里叶系数边界。该技术还可扩展到稀疏及非二分情形（至少当所考虑的子图足够小时）。我们相信这些方法可能有助于解决相关检测问题中的开放性问题。