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

## TL;DR

This paper extends the method of moments for cryo-EM ab initio modeling to handle unknown, non-uniform viewing angle distributions, simplifying the statistical requirements and proposing algorithms for structure reconstruction.

## Contribution

It removes the uniformity assumption in Kam's method, demonstrating that first and second moments suffice for non-uniform distributions and developing algorithms for both known and unknown distributions.

## Key findings

- First and second moments are sufficient for non-uniform distributions.
- Efficient algorithms are proposed for known distribution cases.
- Non-convex optimization methods can recover structure and distribution in unknown cases.

## Abstract

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.