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：我们研究了二分随机几何图（RGGs）中隐几何结构可检测性的信息论相变问题，其中隐向量为高斯d维向量，且仅部分边携带由独立同分布Bern(q)条目构成的随机掩码所确定的隐信息。对于任意固定的边密度p∈(0,1)，我们确定了该问题关于d和q的本质上紧的阈值。结果表明，若掩码预先已知，则检测问题相较于掩码隐藏的情形显著简化。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，推导出相应的界。所得界适用于比先前研究更大的子图规模，从而能够获得紧的信息论界，而先前工作中的界仅能从低次多项式角度给出下界。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。我们的界进一步改进了Bangachev与Bresler [STOC'24] 近期在稠密二分情形下给出的随机几何图傅里叶系数界。该技术亦可扩展至更稀疏及非二分场景，至少当所考虑的子图足够小时成立。我们进一步认为，这些技术可能有助于解决相关检测问题中尚未解决的理论难题。