Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The reflection invariant bispectrum: signal recovery in the dihedral model

Dan Edidin, Josh Katz

TL;DR

This work proves that, in the dihedral MRA model, the orbit of a generic signal under the $n$-dimensional standard representation of the $2n$-element dihedral group is uniquely determined by invariant tensors of degree at most three, yielding a sample complexity of $N=\omega(\sigma^6)$ under uniform group sampling. The authors develop a two-phase approach: first show that degree-3 polynomial invariants separate generic real orbits by leveraging the Fourier/$O(2)$-invariant structure, then extend to complex vectors by showing non-real vectors cannot share these invariants with a generic real vector. They provide detailed analysis across $SO(2)$, $O(2)$, and dihedral representations, including special handling for small or exceptional cases, and demonstrate that a simple third-moment optimization can recover the signal from noisy observations in practice, despite the lack of a practical frequency-marching solution. The results advance understanding of non-abelian MRA, offer concrete recovery guarantees for the dihedral model, and highlight both theoretical and computational avenues for non-abelian group invariants in signal processing. Practical impact arises in areas like cryo-EM where non-abelian symmetries better model the measurement process, and the work includes public code to reproduce experiments.

Abstract

We study the problem of signal recovery in the dihedral multi-reference alignment (MRA) model, where a signal is observed under random actions of the dihedral group and corrupted by additive noise. While previous has shown that cyclic invariants of degree three (the bispectrum) suffice to recover generic signals up to circular shift, the dihedral setting introduces new challenges due to the groups non-abelian structure. In particular reflections prevent the diagonalization of the third moment tensor in the Fourier basis, making classical bispectrum techniques inapplicable. In this work we prove that the orbit of the generic signal in the $n$-dimensional standard representation of the then $2n$-element dihedral group $D_{n}$ is uniquely determined by invariant tensors of degree at most three. This resolves an open question in the literature and establishes that the sample complexity for dihedral MRA with uniform distribution is $ω(σ^6)$ matching the cyclic case. While frequency marching becomes computationally impractical in the dihedral setting, we show numerically that a simple optimization algorithm reliably recovers the signal from third order moments, even with random initialization.

The reflection invariant bispectrum: signal recovery in the dihedral model

TL;DR

This work proves that, in the dihedral MRA model, the orbit of a generic signal under the -dimensional standard representation of the -element dihedral group is uniquely determined by invariant tensors of degree at most three, yielding a sample complexity of under uniform group sampling. The authors develop a two-phase approach: first show that degree-3 polynomial invariants separate generic real orbits by leveraging the Fourier/-invariant structure, then extend to complex vectors by showing non-real vectors cannot share these invariants with a generic real vector. They provide detailed analysis across , , and dihedral representations, including special handling for small or exceptional cases, and demonstrate that a simple third-moment optimization can recover the signal from noisy observations in practice, despite the lack of a practical frequency-marching solution. The results advance understanding of non-abelian MRA, offer concrete recovery guarantees for the dihedral model, and highlight both theoretical and computational avenues for non-abelian group invariants in signal processing. Practical impact arises in areas like cryo-EM where non-abelian symmetries better model the measurement process, and the work includes public code to reproduce experiments.

Abstract

We study the problem of signal recovery in the dihedral multi-reference alignment (MRA) model, where a signal is observed under random actions of the dihedral group and corrupted by additive noise. While previous has shown that cyclic invariants of degree three (the bispectrum) suffice to recover generic signals up to circular shift, the dihedral setting introduces new challenges due to the groups non-abelian structure. In particular reflections prevent the diagonalization of the third moment tensor in the Fourier basis, making classical bispectrum techniques inapplicable. In this work we prove that the orbit of the generic signal in the -dimensional standard representation of the then -element dihedral group is uniquely determined by invariant tensors of degree at most three. This resolves an open question in the literature and establishes that the sample complexity for dihedral MRA with uniform distribution is matching the cyclic case. While frequency marching becomes computationally impractical in the dihedral setting, we show numerically that a simple optimization algorithm reliably recovers the signal from third order moments, even with random initialization.
Paper Structure (18 sections, 15 theorems, 46 equations, 6 figures)

This paper contains 18 sections, 15 theorems, 46 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Invariant tensors
  4. The standard representation of the dihedral group
  5. Outline of the proof
  6. Proof of theorem \ref{['thm.dihedral']} for real vectors
  7. $SO(2)$ and $O(2)$-invariants
  8. Proof for $SO(2)$-invariants
  9. $O(2)$ Invariants
  10. Recovering real vectors from cyclic invariants of low degree
  11. Dihedral invariants
  12. Separation of complex dihedral orbits
  13. Cyclic group
  14. Dihedral group
  15. Open questions
  16. ...and 3 more sections

Key Result

Corollary 1.1

Consider the multi-reference alignment model eq:mra where the $g_i$ are drawn uniformly from $D_n$ and $x \in \mathbb{C}^n$ is a generic vector. Then the minimal number of observations $N$ required for accurate recovery of $x$ is $N/\sigma^6\to\infty$.

Figures (6)

  • Figure 1: An example of the action of the dihedral group. From left to right: the original signal, the signal after a circular shift, and the signal after a shift followed by reflection. The MRA problem \ref{['eq:mra']} entails estimating a signal, up to a global circular shift and reflection, from multiple noisy copies of the signal acted upon by random elements of the dihedral group.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (41)

  • Corollary 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 31 more