Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detecting Correlation between Multiple Unlabeled Gaussian Networks

Taha Ameen, Bruce Hajek

TL;DR

This paper addresses the problem of detecting correlation among m unlabeled graphs with Gaussian edge weights, where correlation is mediated by an unknown latent permutation. It extends the two-graph detection framework to general m by analyzing a generalized likelihood ratio statistic that maximizes over permutation profiles. The authors establish a sufficient condition for strong detection, rho^2 ≥ (8/m) log n/(n−1), and a necessary condition for weak detection, rho^2 ≤ ((4)/(m−1) − ε) log n/n, revealing an interval where detection becomes feasible only when more than two graphs are available. A Gaussian Hanson-Wright variant underpins the analysis, and the results demonstrate how additional graphs reduce the signal requirement, while also highlighting gaps to a sharp threshold and outlining paths for tightening thresholds in future work.

Abstract

This paper studies the hypothesis testing problem to determine whether m > 2 unlabeled graphs with Gaussian edge weights are correlated under a latent permutation. Previously, a sharp detection threshold for the correlation parameter ρwas established by Wu, Xu and Yu for this problem when m = 2. Presently, their result is leveraged to derive necessary and sufficient conditions for general m. In doing so, an interval for ρis uncovered for which detection is impossible using 2 graphs alone but becomes possible with m > 2 graphs.

Detecting Correlation between Multiple Unlabeled Gaussian Networks

TL;DR

This paper addresses the problem of detecting correlation among m unlabeled graphs with Gaussian edge weights, where correlation is mediated by an unknown latent permutation. It extends the two-graph detection framework to general m by analyzing a generalized likelihood ratio statistic that maximizes over permutation profiles. The authors establish a sufficient condition for strong detection, rho^2 ≥ (8/m) log n/(n−1), and a necessary condition for weak detection, rho^2 ≤ ((4)/(m−1) − ε) log n/n, revealing an interval where detection becomes feasible only when more than two graphs are available. A Gaussian Hanson-Wright variant underpins the analysis, and the results demonstrate how additional graphs reduce the signal requirement, while also highlighting gaps to a sharp threshold and outlining paths for tightening thresholds in future work.

Abstract

This paper studies the hypothesis testing problem to determine whether m > 2 unlabeled graphs with Gaussian edge weights are correlated under a latent permutation. Previously, a sharp detection threshold for the correlation parameter ρwas established by Wu, Xu and Yu for this problem when m = 2. Presently, their result is leveraged to derive necessary and sufficient conditions for general m. In doing so, an interval for ρis uncovered for which detection is impossible using 2 graphs alone but becomes possible with m > 2 graphs.

Paper Structure

This paper contains 10 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Preliminaries
  5. Test Statistic and Main Results
  6. A variation of Hanson-Wright inequality for Gaussians
  7. Achievable detection: Proof of Theorem \ref{['thm: positive']}
  8. Impossible detection: Proof of Theorem \ref{['thm: negative']}
  9. Appendix: Proof of Proposition \ref{['prop:HWGaussian']}
  10. Appendix: Proof of lemmas about total variation distance

Key Result

Theorem 2

Suppose that There exists a threshold $\tau$ for which the generalized likelihood ratio test based on $T$ achieves strong detection, i.e. $\mathsf{P}\left( T < \tau \right) + \mathsf{Q}\left( T \geq \tau \right) = o(1).$

Theorems & Definitions (13)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 3 more