Detecting Correlated Gaussian Databases

TL;DR

This paper investigates the detectability of correlation between two Gaussian databases with unknown row permutations, establishing thresholds where detection is possible or impossible, and comparing it to the difficulty of recovering the permutation.

Contribution

The paper introduces a simple detection test for correlated Gaussian databases and characterizes the thresholds for detection feasibility, revealing scenarios where detection is easier than permutation recovery.

Findings

Detection achievable when $ ho^2 oughly 1/d$

Detection impossible below $ ho^2 oughly 1/(d\sqrt{n})$

Detection can be easier than recovery under certain parameters

Abstract

This paper considers the problem of detecting whether two databases, each consisting of $n$ users with $d$ Gaussian features, are correlated. Under the null hypothesis, the databases are independent. Under the alternate hypothesis, the features are correlated across databases, under an unknown row permutation. A simple test is developed to show that detection is achievable above $\rho^2 \approx \frac{1}{d}$. For the converse, the truncated second moment method is used to establish that detection is impossible below roughly $\rho^2 \approx \frac{1}{d\sqrt{n}}$. These results are compared to the corresponding recovery problem, where the goal is to decode the row permutation, and a converse bound of roughly $\rho^2 \approx 1 - n^{-4/d}$ has been previously shown. For certain choices of parameters, the detection achievability bound outperforms this recovery converse bound, demonstrating that detection can be easier than recovery in this scenario.