Improved Computational Lower Bound of Estimation for Multi-Frequency Group Synchronization

TL;DR

This paper investigates the limits of efficient algorithms for multi-frequency group synchronization, revealing a computational barrier that emerges when the number of frequencies grows large, extending previous results to more frequencies.

Contribution

It extends the understanding of computational limits in multi-frequency synchronization, showing spectral methods are optimal under low-degree heuristics for a broader range of frequencies.

Findings

Spectral methods are optimal among polynomial-time algorithms for large frequency counts.

A statistical-to-computational gap exists when the number of frequencies is sufficiently large.

The results extend prior work from fixed to growing number of frequencies.

Abstract

We study the computational phase transition in a multi-frequency group synchronization problem, where pairwise relative measurements of group elements are observed across multiple frequency channels and corrupted by Gaussian noise. Using the framework of \emph{low-degree polynomial algorithms}, we analyze the task of estimating the structured signal in such observations. We show that, assuming the low-degree heuristic, in synchronization models over the circle group $\mathsf{SO}(2)$, a simple spectral method is computationally optimal among all polynomial-time estimators when the number of frequencies satisfies $L=n^{o(1)}$. This significantly extends prior work \cite{KBK24+}, which only applied to a fixed constant number of frequencies. Together with known upper bounds on the statistical threshold \cite{PWBM18a}, our results establish the existence of a \emph{statistical-to-computational gap} in this model when the number of frequencies is sufficiently large.