## Abstract

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 ρ^{2}d → 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^{T}Y. 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 ρ^{2}d = o(1), as n → ∞. These results close significant gaps in current recent related studies.

Original language | English |
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Pages (from-to) | 9246-9266 |

Number of pages | 21 |

Journal | Proceedings of Machine Learning Research |

Volume | 202 |

State | Published - 2023 |

Event | 40th International Conference on Machine Learning, ICML 2023 - Honolulu, United States Duration: 23 Jul 2023 → 29 Jul 2023 |