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：我们研究二部随机几何图（RGG）中潜几何可检测性的信息论相变，其中高斯d维潜向量作为节点嵌入，但仅部分边携带潜信息，该信息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)，我们确定了该问题关于d和q的本质紧阈值。结果表明，与掩码未知情形相比，当掩码先验已知时检测问题显著简化。我们的分析建立在一个新颖的傅里叶分析框架之上，该框架通过近似特征函数的适当幂级数引发的相消来约束高斯随机几何图中的带符号子图计数。所得约束适用于比以往工作更大的子图，从而获得紧的信息论阈值，而此前工作仅通过低次多项式视角得到下界。由此我们识别出最优信息论阈值并排除了计算-统计差距。我们的约束进一步改进了Bangachev与Bresler[STOC'24]近期在稠密二部情形下给出的随机几何图傅里叶系数界。该技术还可推广至稀疏及非二部设置（至少对所考虑子图充分小的情况）。我们相信这些方法可能有助于解决相关检测问题的公开难题。