Projected multi-reference alignment

2026-05-25

AI Summary

This work addresses the problem of recovering the dihedral orbit of a signal from noisy observations subjected to random cyclic shifts, reflections, and a fixed linear projection that discards orientation information. By analyzing the first through third-order moments of the observations in Fourier cosine coordinates, the method sequentially recovers the mean, spectral magnitudes, and phase couplings. The key theoretical contribution is establishing the equivalence—on the level of moments—between projected multi-reference alignment (MRA) and dihedral MRA, and proving that third-order moments uniquely determine the generic signal orbit. Building on this insight, two efficient recovery algorithms are proposed: an expectation–maximization scheme and a direct moment-matching optimization. Both approaches achieve the optimal sample complexity \( n \gtrsim \sigma^6 \) for consistent estimation under high noise, as confirmed by theoretical analysis and numerical experiments.

Abstract

Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation--maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim σ^6$, where $n$ is the number of observations and $σ^2$ is the noise variance.

Problem

• projected multi-reference alignment
• dihedral orbit
• structural biology
• reflection symmetry
• signal recovery

Innovation

• projected multi-reference alignment
• dihedral orbit recovery
• moment-based reconstruction
• Fourier-cosine coordinates
• phase-coupling bispectrum