Estimating the tails of the spectrum of the Hessian of the log-likelihood for \textit{ab-initio} single-particle reconstruction in electron cryomicroscopy

TL;DR

This paper introduces a method to estimate the smallest eigenvalues and eigenvectors of the Hessian of the log-likelihood in cryo-EM single-particle reconstruction, providing insights into solution sensitivity and reliability.

Contribution

It presents a novel strategy to estimate the Hessian's softest modes by reformulating the log-likelihood as a 3D integral, applicable in low-noise and practical scenarios.

Findings

Enables sensitivity analysis of reconstructed volumes.

Identifies trustworthy and unreliable regions in the volume.

Applicable in noise-marginalized scenarios.

Abstract

Electron cryomicroscopy (cryo-EM) is a technique in structural biology used to reconstruct accurate volumetric maps of molecules. One step of the cryo-EM pipeline involves solving an inverse-problem. This inverse-problem, referred to as \textit{ab-initio} single-particle reconstruction, takes as input a collection of 2d-images -- each a projection of a molecule from an unknown viewing-angle -- and attempts to reconstruct the 3d-volume representing the underlying molecular density. Most methods for solving this inverse-problem search for a solution which optimizes a posterior likelihood of generating the observed image-data, given the reconstructed volume. Within this framework, it is natural to study the Hessian of the log-likelihood: the eigenvectors and eigenvalues of the Hessian determine how the likelihood changes with respect to perturbations in the solution, and can give insight into the sensitivity of the solution to aspects of the input. In this paper we describe a simple strategy for estimating the smallest eigenvalues and eigenvectors (i.e., the `softest modes') of the Hessian of the log-likelihood for the \textit{ab-initio} single-particle reconstruction problem. This strategy involves rewriting the log-likelihood as a 3d-integral. This interpretation holds in the low-noise limit, as well as in many practical scenarios which allow for noise-marginalization. Once we have estimated the softest modes, we can use them to perform many kinds of sensitivity analysis. For example, we can determine which parts of the reconstructed volume are trustworthy, and which are unreliable, and how this unreliability might depend on the data-set and the imaging parameters. We believe that this kind of analysis can be used alongside more traditional strategies for sensitivity analysis, as well as in other applications, such as free-energy estimation.