# Single-particle cryo-electron microscopy: Mathematical theory, computational challenges, and opportunities

## TL;DR

This paper reviews the mathematical foundations, computational challenges, and recent theoretical advances in cryo-electron microscopy, emphasizing the integration of diverse computational tools to reconstruct 3D molecular structures from noisy, large-scale data.

## Contribution

It introduces statistical models for cryo-EM data and discusses recent mathematical results, highlighting the interdisciplinary approaches and theoretical insights into the reconstruction process.

## Key findings

- Development of statistical models for cryo-EM data
- Recent mathematical results relating to group and invariant theory
- Insights into computational challenges with large datasets

## Abstract

In recent years, an abundance of new molecular structures have been elucidated using cryo-electron microscopy (cryo-EM), largely due to advances in hardware technology and data processing techniques. Owing to these new exciting developments, cryo-EM was selected by Nature Methods as Method of the Year 2015, and the Nobel Prize in Chemistry 2017 was awarded to three pioneers in the field. The main goal of this article is to introduce the challenging and exciting computational tasks involved in reconstructing 3-D molecular structures by cryo-EM. Determining molecular structures requires a wide range of computational tools in a variety of fields, including signal processing, estimation and detection theory, high-dimensional statistics, convex and non-convex optimization, spectral algorithms, dimensionality reduction, and machine learning. The tools from these fields must be adapted to work under exceptionally challenging conditions, including extreme noise levels, the presence of missing data, and massively large datasets as large as several Terabytes. In addition, we present two statistical models: multi-reference alignment and multi-target detection, that abstract away much of the intricacies of cryo-EM, while retaining some of its essential features. Based on these abstractions, we discuss some recent intriguing results in the mathematical theory of cryo-EM, and delineate relations with group theory, invariant theory, and information theory.