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.
Categories: cs.IT cs.LG math.IT math.ST stat.TH