# Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM

arXiv: https://arxiv.org/abs/2107.01305 · HTML: https://mlsanity.com/paper/2107.01305/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2021.

## Built on by (most cited first, 6 of 6)

- [Dictionary Learning under Symmetries via Group Representations](https://mlsanity.com/paper/2305.19557/) (2023)
- [Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling](https://mlsanity.com/paper/2507.20459/) (2025)
- [Minimax-optimal estimation for sparse multi-reference alignment with collision-free signals](https://mlsanity.com/paper/2312.07839/) (2023)
- [If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics](https://mlsanity.com/paper/2510.09902/) (2025)
- [Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM](https://mlsanity.com/paper/2511.07438/) (2025)
- [Decoy and disclosure radii of invariant shape descriptors](https://mlsanity.com/paper/2610.08410/) (2026)

## Abstract (arXiv)

Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a high noise regime, where samples are observed after random rotation and possible linear projection of the function domain. We describe a stratification of the Fisher information eigenvalues according to transcendence degrees of graded pieces of the algebra of group invariants, and we relate critical points of the log-likelihood landscape to a sequence of moment optimization problems, extending previous results for a discrete rotation group without projections. We then compute the transcendence degrees and forms of these optimization problems for several examples of function estimation under SO(2) and SO(3) rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Perry, Weed, and Wein. We affirmatively resolve conjectures that 3rd-order moments are sufficient to locally identify a generic signal up to its rotational orbit in these examples. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise-scalings of the Fisher information eigenvalues conform with our theoretical predictions over a range of SNR, in a model of SO(3) rotations without projections.