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

Zhangsong Li

TL;DR

This work analyzes estimation in multi-frequency angular synchronization over SO(2) with L frequency channels under Gaussian noise. It leverages the low-degree polynomial framework to show that, for L = n^{o(1)}, a simple spectral method is computationally optimal among polynomial-time estimators, extending prior constant-L results and confirming a statistical-to-computational gap as L grows. Under the low-degree conjecture, when lambda < 1 and L = n^{o(1)}, any estimator running in time n^{o(1)} cannot achieve weak recovery, establishing a concrete computational barrier. The results thus clarify the limits of efficient algorithms in multi-frequency synchronization and justify the observed spectral threshold as a robust computational boundary in this regime.

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.

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

TL;DR

This work analyzes estimation in multi-frequency angular synchronization over SO(2) with L frequency channels under Gaussian noise. It leverages the low-degree polynomial framework to show that, for L = n^{o(1)}, a simple spectral method is computationally optimal among polynomial-time estimators, extending prior constant-L results and confirming a statistical-to-computational gap as L grows. Under the low-degree conjecture, when lambda < 1 and L = n^{o(1)}, any estimator running in time n^{o(1)} cannot achieve weak recovery, establishing a concrete computational barrier. The results thus clarify the limits of efficient algorithms in multi-frequency synchronization and justify the observed spectral threshold as a robust computational boundary in this regime.

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 , a simple spectral method is computationally optimal among all polynomial-time estimators when the number of frequencies satisfies . 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.
Paper Structure (11 sections, 8 theorems, 76 equations)

This paper contains 11 sections, 8 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Group Synchronization
  3. Multi-frequency group synchronization
  4. Overview of proof strategy
  5. Notation
  6. Average-case complexity from low-degree polynomials
  7. Low-degree advantage and low-degree heuristic
  8. Algorithmic contiguity from low-degree heuristic
  9. Proof of Theorem \ref{['MAIN-THM-informal']}
  10. Proof of Lemma \ref{['lem-bound-low-deg-Adv']}
  11. Proof of Lemma \ref{['lem-reduce-to-one-sided-test']}

Key Result

Theorem 1.5

Assuming the low-degree conjecture (see Section subsec:low-degree-conjecture for details), when $L=n^{o(1)}$ and $\lambda<1$, any algorithm for weak recovery requires runtime at least $\exp(n^{\Omega(1)})$.

Theorems & Definitions (25)

  • Definition 1.1: Angular synchronization
  • Definition 1.2: Multi-frequency angular synchronization
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1: Strong/weak detection
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • ...and 15 more