# An accelerated expectation-maximization algorithm for multi-reference alignment

arXiv: https://arxiv.org/abs/2105.07372 · HTML: https://mlsanity.com/paper/2105.07372/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 38 references that are in the map.
- First posted on arXiv 16 May 2021.

## Builds on (most cited first, 10 of 10)

- [Nonconvex phase synchronization](https://mlsanity.com/paper/1601.06114/) (2016)
- [Message-passing algorithms for synchronization problems over compact groups](https://mlsanity.com/paper/1610.04583/) (2016)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [Multireference Alignment using Semidefinite Programming](https://mlsanity.com/paper/1308.5256/) (2013)
- [Non-unique games over compact groups and orientation estimation in cryo-EM](https://mlsanity.com/paper/1505.03840/) (2015)
- [Multi-reference alignment in high dimensions: sample complexity and phase transition](https://mlsanity.com/paper/2007.11482/) (2020)
- [The generalized method of moments for multi-reference alignment](https://mlsanity.com/paper/2103.02215/) (2021)
- [Heterogeneous multireference alignment for images with application to 2-D classification in single particle reconstruction](https://mlsanity.com/paper/1811.10382/) (2018)
- [Super-resolution multi-reference alignment](https://mlsanity.com/paper/2006.15354/) (2020)

## Built on by (most cited first, 1 of 1)

- [Unrolled algorithms for group synchronization](https://mlsanity.com/paper/2207.09418/) (2022)

## Abstract (arXiv)

The multi-reference alignment (MRA) problem entails estimating an image from multiple noisy and rotated copies of itself. If the noise level is low, one can reconstruct the image by estimating the missing rotations, aligning the images, and averaging out the noise. While accurate rotation estimation is impossible if the noise level is high, the rotations can still be approximated, and thus can provide indispensable information. In particular, learning the approximation error can be harnessed for efficient image estimation. In this paper, we propose a new computational framework, called Synch-EM, that consists of angular synchronization followed by expectation-maximization (EM). The synchronization step results in a concentrated distribution of rotations; this distribution is learned and then incorporated into the EM as a Bayesian prior. The learned distribution also dramatically reduces the search space, and thus the computational load, of the EM iterations. We show by extensive numerical experiments that the proposed framework can significantly accelerate EM for MRA in high noise levels, occasionally by a few orders of magnitude, without degrading the reconstruction quality.