In this paper, we investigate the problem of deciding whether two standard normal random vectors $\mathsf{X}\in\mathbb{R}^{n}$ and $\mathsf{Y}\in\mathbb{R}^{n}$ are correlated or not. This is formulated as a hypothesis testing problem, where under the null hypothesis, these vectors are statistically independent, while under the alternative, $\mathsf{X}$ and a randomly and uniformly permuted version of $\mathsf{Y}$, are correlated with correlation $\rho$. We analyze the thresholds at which optimal testing is information-theoretically impossible and possible, as a function of $n$ and $\rho$. To derive our information-theoretic lower bounds, we develop a novel technique for evaluating the second moment of the likelihood ratio using an orthogonal polynomials expansion, which among other things, reveals a surprising connection to integer partition functions. We also study a multi-dimensional generalization of the above setting, where rather than two vectors we observe two databases/matrices, and furthermore allow for partial correlations between these two.

翻译：本文研究判断两个标准正态随机向量$\mathsf{X}\in\mathbb{R}^{n}$与$\mathsf{Y}\in\mathbb{R}^{n}$是否相关的问题。该问题被建模为假设检验：在原假设下，两向量统计独立；在备择假设下，$\mathsf{X}$与$\mathsf{Y}$的随机均匀置换版本以相关系数$\rho$相关。我们分析了最优检验在信息论意义上可行与不可行的阈值，该阈值是$n$与$\rho$的函数。为推导信息论下界，我们发展了一种新技术，通过正交多项式展开计算似然比的二阶矩，该方法揭示了与整数分拆函数之间令人惊奇的关联。此外，我们还研究了上述设定在高维情形下的推广：不再局限于两个向量，而是观测两个数据库/矩阵，并允许两者之间存在部分相关性。