Cryo-electron microscopy (cryo-EM) has emerged as a powerful technique for resolving the three-dimensional structures of macromolecules. A key challenge in cryo-EM is characterizing continuous heterogeneity, where molecules adopt a continuum of conformational states. Covariance-based methods offer a principled approach to modeling structural variability. However, estimating the covariance matrix efficiently remains a challenging computational task. In this paper, we present SOLVAR (Stochastic Optimization for Low-rank Variability Analysis), which leverages a low-rank assumption on the covariance matrix to provide a tractable estimator for its principal components, despite the apparently prohibitive large size of the covariance matrix. Under this low-rank assumption, our estimator can be formulated as an optimization problem that can be solved quickly and accurately. Moreover, our framework enables refinement of the poses of the input particle images, a capability absent from most heterogeneity-analysis methods, and all covariance-based methods. Numerical experiments on both synthetic and experimental datasets demonstrate that the algorithm accurately captures dominant components of variability while maintaining computational efficiency. SOLVAR achieves state-of-the-art performance across multiple datasets in a recent heterogeneity benchmark. The code of the algorithm is freely available at https://github.com/RoeyYadgar/SOLVAR.

翻译：冷冻电子显微镜（cryo-EM）已成为解析大分子三维结构的重要技术。冷冻电镜分析中的一个关键挑战在于表征连续异质性，即分子呈现连续构象状态的情况。基于协方差的方法为建模结构变异性提供了理论框架，然而高效估计协方差矩阵仍是一项计算难题。本文提出SOLVAR（面向低秩变异性分析的随机优化方法），该方法利用协方差矩阵的低秩假设，为其主成分提供可处理的估计量，尽管协方差矩阵的表观维度极大。在此低秩假设下，我们的估计量可表述为一个能快速精确求解的优化问题。此外，本框架能够对输入颗粒图像的姿态进行优化，这是大多数异质性分析方法及所有基于协方差的方法所不具备的功能。在合成与实验数据集上的数值实验表明，该算法在保持计算效率的同时，能准确捕捉变异性的主要成分。在近期异质性基准测试中，SOLVAR在多个数据集上实现了最先进的性能。算法代码已公开于 https://github.com/RoeyYadgar/SOLVAR。