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.

研究标准正态随机向量之间是否存在相关性的假设检验问题，提出了信息理论下界的评估方法，同时探索了多维情况下观察两个数据库 / 矩阵之间的部分相关性。