Dihedral multi-reference alignment

TL;DR

This paper investigates the dihedral multi-reference alignment problem, establishing conditions for unique orbit recovery from noisy observations and proposing computational methods for signal estimation.

Contribution

It provides the first theoretical results for non-abelian groups with non-uniform distributions and introduces three numerical frameworks for signal estimation.

Findings

Unique orbit recovery from second moments in high noise

Optimal estimation rate proportional to noise variance squared

Development of three computational estimation frameworks

Abstract

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.