Detection of Correlated Random Vectors

In this paper, we investigate the problem of deciding whether two standard normal random vectors $\textsf {X}\in \mathbb {R}^{n}$ and $\textsf {Y}\in \mathbb {R}^{n}$ 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, $\textsf {X}$ and a randomly and uniformly permuted version of $\textsf {Y}$ , 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.