# Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles

arXiv: https://arxiv.org/abs/1907.05377 · HTML: https://mlsanity.com/paper/1907.05377/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2019.

## Built on by (most cited first, 8 of 8)

- [Deep Generative Modeling for Volume Reconstruction in Cryo-Electron Microscopy](https://mlsanity.com/paper/2201.02867/) (2022)
- [Tensor Moments of Gaussian Mixture Models: Theory and Applications](https://mlsanity.com/paper/2202.06930/) (2022)
- [Learning to recover orientations from projections in single-particle cryo-EM](https://mlsanity.com/paper/2104.06237/) (2021)
- [Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown-View Tomography](https://mlsanity.com/paper/2207.02985/) (2022)
- [Multi-target detection with rotations](https://mlsanity.com/paper/2101.07709/) (2021)
- [The G-invariant graph Laplacian](https://mlsanity.com/paper/2303.17001/) (2023)
- [Deep Neural-network Prior for Orbit Recovery from Method of Moments](https://mlsanity.com/paper/2304.14604/) (2023)
- [Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM](https://mlsanity.com/paper/2511.07438/) (2025)

## Abstract (arXiv)

Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an increasingly popular technique for determining the 3-D structure of a molecule from several noisy 2-D projections images taken at unknown viewing angles. Most reconstruction algorithms require a low-resolution initialization for the 3-D structure, which is the goal of ab initio modeling. Suggested by Zvi Kam in 1980, the method of moments (MoM) offers one approach, wherein low-order statistics of the 2-D images are computed and a 3-D structure is estimated by solving a system of polynomial equations. Unfortunately, Kam's method suffers from restrictive assumptions, most notably that viewing angles should be distributed uniformly. Often unrealistic, uniformity entails the computation of higher-order correlations, as in this case first and second moments fail to determine the 3-D structure. In the present paper, we remove this hypothesis, by permitting an unknown, non-uniform distribution of viewing angles in MoM. Perhaps surprisingly, we show that this case is statistically easier than the uniform case, as now first and second moments generically suffice to determine low-resolution expansions of the molecule. In the idealized setting of a known, non-uniform distribution, we find an efficient provable algorithm inverting first and second moments. For unknown, non-uniform distributions, we use non-convex optimization methods to solve for both the molecule and distribution.