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Electron cryomicroscopy (cryo-EM) is an imaging technique widely used instructural biology to determine the three-dimensional structure of biologicalmolecules from noisy two-dimensional projections with unknown orientations. Asthe typical pipeline involves processing large amounts of data, efficientalgorithms are crucial for fast and reliable results. The stochastic gradientdescent (SGD) algorithm has been used to improve the speed of ab initioreconstruction, which results in a first, low-resolution estimation of thevolume representing the molecule of interest, but has yet to be appliedsuccessfully in the high-resolution regime, where expectation-maximizationalgorithms achieve state-of-the-art results, at a high computational cost. Inthis article, we investigate the conditioning of the optimization problem andshow that the large condition number prevents the successful application ofgradient descent-based methods at high resolution. Our results include atheoretical analysis of the condition number of the optimization problem in asimplified setting where the individual projection directions are known, analgorithm based on computing a diagonal preconditioner using Hutchinson'sdiagonal estimator, and numerical experiments showing the improvement in theconvergence speed when using the estimated preconditioner with SGD. Thepreconditioned SGD approach can potentially enable a simple and unifiedapproach to ab initio reconstruction and high-resolution refinement with fasterconvergence speed and higher flexibility, and our results are a promising stepin this direction.

Efficient high-resolution refinement in cryo-EM with stochastic gradient descent