Phase Transitions in the Detection of Correlated Databases

• URL: http://arxiv.org/abs/2302.03380v2
• Date: Thu, 4 May 2023 06:19:51 GMT
• Title: Phase Transitions in the Detection of Correlated Databases
• Authors: Dor Elimelech and Wasim Huleihel
• Abstract summary: We study the problem of detecting the correlation between two Gaussian databases $mathsfXinmathbbRntimes d$ and $mathsfYntimes d$, each composed of $n$ users with $d$ features. This problem is relevant in the analysis of social media, computational biology, etc.
• Score: 12.010807505655238
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of detecting the correlation between two Gaussian databases $\mathsf{X}\in\mathbb{R}^{n\times d}$ and $\mathsf{Y}^{n\times d}$, 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 $\sigma$ over the set of $n$ users (or, row permutation), such that $\mathsf{X}$ is $\rho$-correlated with $\mathsf{Y}^\sigma$, a permuted version of $\mathsf{Y}$. 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 $\rho^2d\to0$, as $d\to\infty$, 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 $\mathsf{X}^T\mathsf{Y}$. Furthermore, when $d$ is fixed, we prove that strong detection (vanishing error probability) is impossible for any $\rho<\rho^\star$, where $\rho^\star$ is an explicit function of $d$, while weak detection is again impossible as long as $\rho^2d\to0$. These results close significant gaps in current recent related studies.

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