Testing Dependency of Unlabeled Databases

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 $d\to \mathrm{\infty }$, 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.