# Unrolled algorithms for group synchronization

arXiv: https://arxiv.org/abs/2207.09418 · HTML: https://mlsanity.com/paper/2207.09418/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 19 Jul 2022.

## Builds on (most cited first, 27 of 27)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [A Survey of Structure from Motion](https://mlsanity.com/paper/1701.08493/) (2017)
- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [Momentum-Net: Fast and convergent iterative neural network for inverse problems](https://mlsanity.com/paper/1907.11818/) (2019)
- [Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising](https://mlsanity.com/paper/2006.01301/) (2020)
- [Near-optimal bounds for phase synchronization](https://mlsanity.com/paper/1703.06605/) (2017)
- [Nonconvex phase synchronization](https://mlsanity.com/paper/1601.06114/) (2016)
- [Tightness of the maximum likelihood semidefinite relaxation for angular synchronization](https://mlsanity.com/paper/1411.3272/) (2014)
- [Message-passing algorithms for synchronization problems over compact groups](https://mlsanity.com/paper/1610.04583/) (2016)
- [Near-Optimal Performance Bounds for Orthogonal and Permutation Group Synchronization via Spectral Methods](https://mlsanity.com/paper/2008.05341/) (2020)
- [Sync-Rank: Robust Ranking, Constrained Ranking and Rank Aggregation via Eigenvector and Semidefinite Programming Synchronization](https://mlsanity.com/paper/1504.01070/) (2015)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [An Algorithm Unrolling Approach to Deep Image Deblurring](https://mlsanity.com/paper/1902.05399/) (2019)
- [Multireference Alignment using Semidefinite Programming](https://mlsanity.com/paper/1308.5256/) (2013)
- [Deep Unrolled Recovery in Sparse Biological Imaging](https://mlsanity.com/paper/2109.14025/) (2021)
- [Non-unique games over compact groups and orientation estimation in cryo-EM](https://mlsanity.com/paper/1505.03840/) (2015)
- [Alternating Projection, Ptychographic Imaging and Phase Synchronization](https://mlsanity.com/paper/1402.0550/) (2014)
- [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)
- [Synchronization over Cartan motion groups via contraction](https://mlsanity.com/paper/1612.00059/) (2016)
- [Heterogeneous multireference alignment for images with application to 2-D classification in single particle reconstruction](https://mlsanity.com/paper/1811.10382/) (2018)
- [Dihedral multi-reference alignment](https://mlsanity.com/paper/2107.05262/) (2021)
- [Compactification of the Rigid Motions Group in Image Processing](https://mlsanity.com/paper/2106.13505/) (2021)
- [An accelerated expectation-maximization algorithm for multi-reference alignment](https://mlsanity.com/paper/2105.07372/) (2021)

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

- [Robust Angular Synchronization via Directed Graph Neural Networks](https://mlsanity.com/paper/2310.05842/) (2023)

## Abstract (arXiv)

The group synchronization problem involves estimating a collection of group elements from noisy measurements of their pairwise ratios. This task is a key component in many computational problems, including the molecular reconstruction problem in single-particle cryo-electron microscopy (cryo-EM). The standard methods to estimate the group elements are based on iteratively applying linear and non-linear operators, and are not necessarily optimal. Motivated by the structural similarity to deep neural networks, we adopt the concept of algorithm unrolling, where training data is used to optimize the algorithm. We design unrolled algorithms for several group synchronization instances, including synchronization over the group of 3-D rotations: the synchronization problem in cryo-EM. We also apply a similar approach to the multi-reference alignment problem. We show by numerical experiments that the unrolling strategy outperforms existing synchronization algorithms in a wide variety of scenarios.