Computational lower bounds for multi-frequency group synchronization

TL;DR

This paper investigates the computational limits of detecting signals in multi-frequency group synchronization problems with Gaussian noise, revealing a statistical-to-computational gap and establishing the optimality of spectral algorithms under certain conditions.

Contribution

It introduces a complexity-theoretic analysis of multi-frequency group synchronization, demonstrating a computational barrier and identifying the optimality of spectral algorithms under the low-degree conjecture.

Findings

Spectral algorithms are optimal for detection within certain runtime bounds.

A statistical-to-computational gap exists with many frequencies.

The results apply to arbitrary finite groups and the circle group SO(2).

Abstract

We consider a group synchronization problem with multiple frequencies which involves observing pairwise relative measurements of group elements on multiple frequency channels, corrupted by Gaussian noise. We study the computational phase transition in the problem of detecting whether a structured signal is present in such observations by analyzing low-degree polynomial algorithms. We show that, assuming the low-degree conjecture, in synchronization models over arbitrary finite groups as well as over the circle group $SO(2)$, a simple spectral algorithm is optimal among algorithms of runtime $\exp(\tilde{\Omega}(n^{1/3}))$ for detection from an observation including a constant number of frequencies. Combined with an upper bound for the statistical threshold shown in Perry et al., our results indicate the presence of a statistical-to-computational gap in such models with a sufficiently large number of frequencies.