We study the problem of detecting the correlation between two Gaussian databases $\mathsf{X}\in\mathbb{R}^{n\times d}$ and $\mathsf{Y}^{n\times d}$, each composed of $n$ users with $d$ features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation $\sigma$ over the set of $n$ users (or, row permutation), such that $\mathsf{X}$ is $\rho$-correlated with $\mathsf{Y}^\sigma$, a permuted version of $\mathsf{Y}$. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of $n$ and $d$. Specifically, we prove that if $\rho^2d\to0$, as $d\to\infty$, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of $n$. This compliments the performance of a simple test that thresholds the sum all entries of $\mathsf{X}^T\mathsf{Y}$. Furthermore, when $d$ is fixed, we prove that strong detection (vanishing error probability) is impossible for any $\rho<\rho^\star$, where $\rho^\star$ is an explicit function of $d$, while weak detection is again impossible as long as $\rho^2d\to0$. These results close significant gaps in current recent related studies.

翻译：我们研究两个高斯数据库 $\mathsf{X}\in\mathbb{R}^{n\times d}$ 与 $\mathsf{Y}^{n\times d}$ 之间相关性的检测问题，每个数据库包含 $n$ 个用户及 $d$ 个特征。该问题在社交媒体分析、计算生物学等领域具有重要应用。我们将此问题建模为假设检验：在原假设下，这两个数据库统计独立；而在备择假设下，存在一个未知的 $n$ 个用户上的置换 $\sigma$（即行置换），使得 $\mathsf{X}$ 与 $\mathsf{Y}^\sigma$（$\mathsf{Y}$ 经置换后的版本）具有 $\rho$ 相关性。我们确定了最优检验发生相变时的尖锐阈值，该阈值取决于 $n$ 与 $d$ 的渐近机制。具体而言，我们证明：若当 $d\to\infty$ 时 $\rho^2d\to0$，则无论 $n$ 取何值，弱检测（表现略优于随机猜测）在统计上均不可实现。这补充了对 $\mathsf{X}^T\mathsf{Y}$ 所有元素求和进行阈值化这一简单检验的性能分析。此外，当 $d$ 固定时，我们证明：对于任意 $\rho<\rho^\star$（其中 $\rho^\star$ 为 $d$ 的显式函数），强检测（错误概率趋近于零）不可实现；而弱检测在 $\rho^2d\to0$ 条件下同样不可实现。这些结果弥补了近期相关研究中存在的显著空白。