# Dihedral multi-reference alignment

arXiv: https://arxiv.org/abs/2107.05262 · HTML: https://mlsanity.com/paper/2107.05262/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2021.

## Built on by (most cited first, 3 of 3)

- [Group-invariant max filtering](https://mlsanity.com/paper/2205.14039/) (2022)
- [Unrolled algorithms for group synchronization](https://mlsanity.com/paper/2207.09418/) (2022)
- [If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics](https://mlsanity.com/paper/2510.09902/) (2025)

## Abstract (arXiv)

We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from a generic distribution, the orbit of a generic signal is uniquely determined from the second moment of the observations. This implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise. This is the first result of this type for multi-reference alignment over a non-abelian group with a non-uniform distribution of group elements. Based on tools from invariant theory and algebraic geometry, we also delineate conditions for unique orbit recovery for multi-reference alignment models over finite groups (namely, when the dihedral group is replaced by a general finite group) when the group elements are drawn from a generic distribution. Finally, we design and study numerically three computational frameworks for estimating the signal based on group synchronization, expectation-maximization, and the method of moments.