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 $n \gtrsim \sigma^6$, where $n$ is the number of observations and $\sigma^2$ is the noise variance.
| Subjects: | Signal Processing (eess.SP); Information Theory (cs.IT); Statistics Theory (math.ST) |
| Cite as: | arXiv:2605.25533 [eess.SP] |
| (or arXiv:2605.25533v1 [eess.SP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25533 arXiv-issued DOI via DataCite (pending registration) |
From: Tamir Bendory [view email]
[v1]
Mon, 25 May 2026 07:39:59 UTC (586 KB)
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