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Computational lower bounds for multi-frequency group synchronization

Anastasia Kireeva, Afonso S. Bandeira, Dmitriy Kunisky

TL;DR

This work establishes low-degree polynomial lower bounds for multi-frequency group synchronization over both the circle group and general finite groups, under a standard conjecture linking low-degree hardness to computational limits. By modeling observations as Gaussian spiked matrices across multiple frequencies and leveraging the Peter–Weyl decomposition, the authors show that with a constant number of frequencies, detection remains computationally hard at signal strengths up to the spectral threshold (e.g., $\lambda_{\max} \le 1$), while a simple PCA test suffices when a single frequency crosses the threshold. The finite-group analysis extends these results via a detailed combinatorial LDLR computation based on representation theory, demonstrating a comparable hardness regime for general $G$ (with $|G|=L$) and highlighting a potential statistical-to-computational gap in such models. These results support AMP-based predictions and clarify when multi-frequency information does not yield computational advantages for detection, while also outlining directions for future work on thresholds, infinite groups, and diverging frequencies. Overall, the paper connects average-case hardness with group-structure-aware multi-frequency observations to characterize computational limits in group synchronization tasks.

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 $\exp(\tildeΩ(n^{1/3}))$ 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.

Computational lower bounds for multi-frequency group synchronization

TL;DR

This work establishes low-degree polynomial lower bounds for multi-frequency group synchronization over both the circle group and general finite groups, under a standard conjecture linking low-degree hardness to computational limits. By modeling observations as Gaussian spiked matrices across multiple frequencies and leveraging the Peter–Weyl decomposition, the authors show that with a constant number of frequencies, detection remains computationally hard at signal strengths up to the spectral threshold (e.g., ), while a simple PCA test suffices when a single frequency crosses the threshold. The finite-group analysis extends these results via a detailed combinatorial LDLR computation based on representation theory, demonstrating a comparable hardness regime for general (with ) and highlighting a potential statistical-to-computational gap in such models. These results support AMP-based predictions and clarify when multi-frequency information does not yield computational advantages for detection, while also outlining directions for future work on thresholds, infinite groups, and diverging frequencies. Overall, the paper connects average-case hardness with group-structure-aware multi-frequency observations to characterize computational limits in group synchronization tasks.

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 , a simple spectral algorithm is optimal among algorithms of runtime 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.
Paper Structure (32 sections, 23 theorems, 127 equations, 1 figure, 1 table)

This paper contains 32 sections, 23 theorems, 127 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Average-case complexity from low-degree polynomials
  4. Statistical distinguishability
  5. Low-degree likelihood ratio (LDLR)
  6. Synchronization over the circle group: main results
  7. Synchronization over the circle group: proofs
  8. Bounding $\mathop{\mathrm{GSynch}}\limits(\mathbb S, L, \lambda)$ by $\mathop{\mathrm{GSynch}}\limits(\mathbb Z_L, \lambda)$
  9. Multinomial expression of $\| L^{\le D}_n \|^2$ for $\mathop{\mathrm{GSynch}}\limits(\mathbb Z_L, \lambda)$
  10. Sketch of combinatorial analysis of $\mathop{\mathrm{GSynch}}\limits(\mathbb Z_L, \lambda)$
  11. Synchronization over finite groups: main results
  12. Short preliminaries on representation theory
  13. Normalized Haar measure
  14. Irreducible representations
  15. Peter-Weyl decomposition
  16. ...and 17 more sections

Key Result

Theorem 1.1

Consider the angular synchronization model eq:intro_ang_def with $L$ frequencies, where $L$ is a constant that does not depend on $n$. If the Low-Degree Conjecture holds (see sec:low-degree-intro), then for any $\lambda \le 1$, any algorithm for strong detection requires runtime at least $\exp(\tild

Figures (1)

  • Figure 1: A schematic illustration of phase transitions in the multi-frequency synchronization model over a finite group of order $L$ (for sufficiently large $L$). $1 / \sqrt{L}$ on the far left is included as it is the "optimistic" though incorrect computational threshold we would expect if independent observations of different frequencies behaved like independent observations of the same frequency. The "possible but hard" regime corresponds to the statistical-to-computational gap in the low-degree sense. In this signal-to-noise regime, strong detection is information-theoretically possible; however, conjecturally, there are no efficient algorithms achieving it.

Theorems & Definitions (59)

  • Definition 1
  • Theorem 1.1: $\mathbb S$-synchronization lower bound; informal
  • Theorem 1.2: Finite group synchronization lower bound; informal
  • Remark 1: Equivalent model: noisy indicators
  • Remark 2
  • Definition 2
  • Lemma 2.1
  • Definition 3
  • Conjecture 1
  • Definition 4
  • ...and 49 more