Phase Transitions in the Detection of Correlated Databases

We study the problem of detecting the correlation between two Gaussian databases X ∈ ℝ^n×d and Y^n×d, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation σ over the set of n users (or, row permutation), such that X is ρ-correlated with Y^σ, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if ρ²d → 0, as d → ∞, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any ρ < ρ^*, where ρ^* is an explicit function of d, while weak detection is again impossible as long as ρ²d = o(1), as n → ∞. These results close significant gaps in current recent related studies.