Projected multi-reference alignment
2026-05-25 • Information Theory
Information Theory
AI summaryⓘ
The authors study a problem where a hidden signal is observed through many noisy samples that are shifted and then projected in a way that loses some orientation information. They show that by looking at certain statistical properties (called moments) of these observations—specifically the first three moments—they can uniquely recover the original signal structure in high noise. They also explain how these moments relate to familiar mathematical objects like Fourier components and phase relations. Their experiments confirm that common algorithms work as predicted when there are enough samples compared to the noise level.
multi-reference alignment (MRA)dihedral orbitmoment analysisFourier transformphase couplingbispectrumprojectionhigh-noise regimeexpectation-maximization (EM)sample complexity
Authors
Amnon Balanov, Josh Katz, Tamir Bendory, Dan Edidin
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.