Testing Dependency of Unlabeled Databases

• URL: http://arxiv.org/abs/2311.05874v1
• Date: Fri, 10 Nov 2023 05:17:03 GMT
• Title: Testing Dependency of Unlabeled Databases
• Authors: Vered Paslev and Wasim Huleihel
• Abstract summary: Two random databases $mathsfXinmathcalXntimes d$ and $mathsfYinmathcalYntimes d$ are statistically dependent or not. We characterize the thresholds at which optimal testing is information-theoretically impossible and possible.
• Score: 5.384630221560811
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $d\to\infty$, 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.

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