A Fast Generative Framework for High-dimensional Posterior Sampling: Application to CMB Delensing

Modeling galaxy-galaxy strong gravitational lenses to infer the brightness of the source galaxy and the mass distribution of the foreground galaxy is computationally challenging, particularly for high-resolution, high signal-to-noise ratio observations. In this regime, high-dimensional representations of both the source and the foreground mass distribution are necessary to model the data down to the noise level. This inference problem has been challenging for both traditional and machine learning-based methods because of its high dimensionality and its non-linearity in the foreground mass distribution. We present a method to generate joint posterior samples of the source galaxy and foreground mass distribution as pixelated images conditioned on observations. The method combines diffusion-based generative modeling and recurrent inference machines. It can model realistic gravitational lensing simulations with background and foreground galaxies drawn from cosmological hydrodynamical simulations down to the noise level.

Accurate posterior estimation is central to scientific inference, as uncertainties determine what can be reliably learned from observational data. While Markov chain Monte Carlo methods provide asymptotic convergence guarantees, they are computationally demanding in high-dimensional settings. Neural network-based generative models for entire discretized 3D fields enable fast amortized inference but often lack convergence guarantees and principled accuracy assessment. Using Hamiltonian Monte Carlo to obtain reference posterior samples, we conduct a controlled field-level evaluation of an implicit generative model (Stochastic Interpolants) and an explicit likelihood-based model (GLOW normalizing flows). This comparison, unavailable in typical applications, enables the detection of posterior geometry failures that standard metrics cannot capture. As a case study, we consider the cosmological inverse problem of inferring cosmic initial conditions from present-day large-scale structure. To match the precision of modern cosmological data, this problem increasingly relies on complex, non-linear, and non-differentiable simulators, which are incompatible with gradient-based inference frameworks. Generative models offer a route to address these challenges, provided their inferred posteriors are reliable. In this work, we show that matching posterior means, marginal distributions, or achieving high cross-correlation does not imply correct uncertainty structure, as revealed by posterior variance fields and sample-based evaluations. Through this work, we aim to raise awareness of the challenges of uncertainty estimation in high-dimensional field-level settings, highlighting the importance of careful design and validation of neural generative approaches for scientific applications.

Markov chain Monte Carlo (MCMC) methods are fundamental to Bayesian computation, but can be computationally intensive, especially in high-dimensional settings. Push-forward generative models, such as generative adversarial networks (GANs), variational auto-encoders and normalising flows offer a computationally efficient alternative for posterior sampling. However, push-forward models are opaque as they lack the modularity of Bayes Theorem, leading to poor generalisation with respect to changes in the likelihood function. In this work, we introduce a novel approach to GAN architecture design by applying deep unfolding to Langevin MCMC algorithms. This paradigm maps fixed-step iterative algorithms onto modular neural networks, yielding architectures that are both flexible and amenable to interpretation. Crucially, our design allows key model parameters to be specified at inference time, offering robustness to changes in the likelihood parameters. We train these unfolded samplers end-to-end using a supervised regularized Wasserstein GAN framework for posterior sampling. Through extensive Bayesian imaging experiments, we demonstrate that our proposed approach achieves high sampling accuracy and excellent computational efficiency, while retaining the physics consistency, adaptability and interpretability of classical MCMC strategies.