Electron cryomicroscopy (cryo-EM) is an imaging technique widely used in structural biology to determine the three-dimensional structure of biological molecules from noisy two-dimensional projections with unknown orientations. As the typical pipeline involves processing large amounts of data, efficient algorithms are crucial for fast and reliable results. The stochastic gradient descent (SGD) algorithm has been used to improve the speed of ab initio reconstruction, which results in a first, low-resolution estimation of the volume representing the molecule of interest, but has yet to be applied successfully in the high-resolution regime, where expectation-maximization algorithms achieve state-of-the-art results, at a high computational cost. In this article, we investigate the conditioning of the optimization problem and show that the large condition number prevents the successful application of gradient descent-based methods at high resolution. Our results include a theoretical analysis of the condition number of the optimization problem in a simplified setting where the individual projection directions are known, an algorithm based on computing a diagonal preconditioner using Hutchinson's diagonal estimator, and numerical experiments showing the improvement in the convergence speed when using the estimated preconditioner with SGD. The preconditioned SGD approach can potentially enable a simple and unified approach to ab initio reconstruction and high-resolution refinement with faster convergence speed and higher flexibility, and our results are a promising step in this direction.

翻译：电子冷冻显微术（cryo-EM）是结构生物学中广泛应用的成像技术，通过噪声干扰的二维投影图像反演未知取向的生物大分子三维结构。由于标准流程涉及处理海量数据，高效算法对实现快速可靠的结果至关重要。随机梯度下降（SGD）算法已被用于提升从头重建速度，可获得目标分子初始的低分辨率体积估计，但其在高分辨率领域尚未成功应用——该领域虽已通过期望最大化算法达到最优结果，却需付出高昂的计算成本。本文通过分析优化问题的条件数，证明在传统梯度下降方法中，较高条件数会阻碍高分辨率下的有效应用。我们取得以下成果：在投影方向已知的简化场景下对优化问题条件数进行理论分析；提出基于Hutchinson对角估计器计算对角预条件子的算法；通过数值实验验证使用该预条件子改进SGD收敛速度的效果。预条件SGD方法有望实现从头重建与高分辨率精化的统一简易框架，兼具更快的收敛速度与更高的灵活性，本研究为此方向提供了关键进展。