Empirical Bayes Flow Matching for Continuous Cryo-EM Heterogeneity

arXiv:2611.05202v1 Announce Type: new Abstract: We address the problem of learning generative priors over high-dimensional latent variables from indirect, noisy observations, with a focus on continuous heterogeneity in cryo-electron microscopy (cryo-EM). In this setting, 3D atomic structures $x \in \mathbb{R}^{3N_a}$ are never directly observed and must be inferred from 2D projection images $y$ generated by a complex, non-linear forward operator. We propose Empirical Bayes Flow Matching (EB-FM), a method that learns a flow matching prior $p_\theta(x)$ directly from observations using a Monte Carlo stochastic approximation Expectation-Maximization framework. EB-FM alternates between an E-step that performs approximate posterior sampling from $p_\theta(x \mid y)$ via guided flow-based generation while maintaining persistent latent estimates through stochastic approximation, and an M-step that updates the flow model using these stabilized latent variables as training targets. This approach eliminates the need for pretraining on clean data and naturally accommodates non-linear forward models. We validate our method on controlled inverse problems using two-moons and MNIST data, and demonstrate its effectiveness on simulated cryo-EM data with known poses, showing that EB-FM can recover continuous distributions of protein conformations directly in atomic-coordinate space.

Abstract:We address the problem of learning generative priors over high-dimensional latent variables from indirect, noisy observations, with a focus on continuous heterogeneity in cryo-electron microscopy (cryo-EM). In this setting, 3D atomic structures $x \in \mathbb{R}^{3N_a}$ are never directly observed and must be inferred from 2D projection images $y$ generated by a complex, non-linear forward operator. We propose Empirical Bayes Flow Matching (EB-FM), a method that learns a flow matching prior $p_\theta(x)$ directly from observations using a Monte Carlo stochastic approximation Expectation-Maximization framework. EB-FM alternates between an E-step that performs approximate posterior sampling from $p_\theta(x \mid y)$ via guided flow-based generation while maintaining persistent latent estimates through stochastic approximation, and an M-step that updates the flow model using these stabilized latent variables as training targets. This approach eliminates the need for pretraining on clean data and naturally accommodates non-linear forward models. We validate our method on controlled inverse problems using two-moons and MNIST data, and demonstrate its effectiveness on simulated cryo-EM data with known poses, showing that EB-FM can recover continuous distributions of protein conformations directly in atomic-coordinate space.

Accepted at NeurIPS 2026. 20 pages, 6 figures, 5 tables. Code: this https URL

Machine Learning (cs.LG); Biomolecules (q-bio.BM)

arXiv:2611.05202 [cs.LG]

(or arXiv:2611.05202v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2611.05202 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Empirical Bayes Flow Matching for Continuous Cryo-EM Heterogeneity

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