In this paper, we investigate the problem of deciding whether two random databases $\mathsf{X}\in\mathcal{X}^{n\times d}$ and $\mathsf{Y}\in\mathcal{Y}^{n\times d}$ are statistically dependent or not. This is formulated as a hypothesis testing problem, where under the null hypothesis, these two databases are statistically independent, while under the alternative, there exists an unknown row permutation $\sigma$, such that $\mathsf{X}$ and $\mathsf{Y}^\sigma$, a permuted version of $\mathsf{Y}$, are statistically dependent with some known joint distribution, but have the same marginal distributions as the null. We characterize the thresholds at which optimal testing is information-theoretically impossible and possible, as a function of $n$, $d$, and some spectral properties of the generative distributions of the datasets. For example, we prove that if a certain function of the eigenvalues of the likelihood function and $d$, is below a certain threshold, as $d\to\infty$, then weak detection (performing slightly better than random guessing) is statistically impossible, no matter what the value of $n$ is. This mimics the performance of an efficient test that thresholds a centered version of the log-likelihood function of the observed matrices. We also analyze the case where $d$ is fixed, for which we derive strong (vanishing error) and weak detection lower and upper bounds.

翻译：本文研究判定两个随机数据库 $\mathsf{X}\in\mathcal{X}^{n\times d}$ 与 $\mathsf{Y}\in\mathcal{Y}^{n\times d}$ 是否具有统计依赖性的问题。该问题被形式化为一个假设检验：在原假设下，这两个数据库统计独立；而在备择假设下，存在一个未知的行置换 $\sigma$，使得 $\mathsf{X}$ 与 $\mathsf{Y}^\sigma$（$\mathsf{Y}$ 的置换版本）在给定联合分布下统计相关，但边缘分布与原假设相同。我们刻画了最优检验在信息论意义上不可能与可能实现的阈值，该阈值取决于 $n$、$d$ 以及数据生成分布的谱特性。例如，我们证明：当似然函数特征值与 $d$ 的某个函数低于特定阈值时（随着 $d\to\infty$），弱检测（略优于随机猜测）在统计上是不可能的——无论 $n$ 取何值。这一结果与基于观测矩阵中心化对数似然函数阈值的有效检验性能相匹配。我们还分析了 $d$ 固定情形，并推导了强检测（误差趋近于零）与弱检测的下界与上界。