Generative Inverse Sampling

• Generative Inverse Sampling is a probabilistic framework that samples from Bayesian posterior distributions using deep generative models as informative priors.
• It integrates methodologies like diffusion, flow-based, and plug-and-play techniques to guide reconstruction while ensuring data fidelity in noisy, underdetermined settings.
• The approach enables uncertainty quantification and diverse solution reconstructions in applications ranging from imaging science to inverse design and scientific data assimilation.

Generative inverse sampling refers to a family of probabilistic and algorithmic frameworks that enable the sampling of solutions from the (approximate) posterior distribution over latent variables or signals, conditioned on noisy and often underdetermined measurements, by leveraging expressive generative models as priors. Bridging model-based Bayesian inference with modern deep generative priors—diffusion models, flows, GANs, and others—generative inverse sampling allows for both uncertainty quantification and high-quality, sample-diverse reconstructions in challenging inverse problems. The recent decade has witnessed a dramatic expansion and unification of generative inverse sampling methodologies across imaging science, physics, engineering, and scientific data assimilation, with significant methodological and empirical advances.

1. Bayesian Inverse Problem Formulation and Posterior Sampling

The canonical generative inverse sampling setup starts from the description of an inverse problem: given observed measurements $y \in \mathbb{R}^m$ generated by an (possibly nonlinear) forward model $\mathcal{A}: \mathbb{R}^n \to \mathbb{R}^m$ acting on an unknown latent variable $x \in \mathbb{R}^n$, with additive noise $n$ (typically $n \sim \mathcal{N}(0, \sigma_y^2 I)$ or non-Gaussian):

$y = \mathcal{A}(x) + n,$

the goal is to sample from the conditional posterior $p(x|y)$, defined via Bayes' theorem as

$p(x|y) \propto p(y|x) \, p(x),$

where $p(x)$ encodes prior knowledge about $x$, generally via a deep generative model, and $p(y|x)$ is the likelihood derived from the forward process and the noise model. These settings appear in applications such as image denoising, super-resolution, inpainting, MRI, tomography, physical system state estimation, and nonlinear inverse design (Bouman et al., 2023, Alkhouri et al., 2024, Andrae et al., 29 Nov 2025, Meng et al., 15 Jun 2025, Lin et al., 30 Jan 2026, Kim et al., 11 Mar 2025, Tanevardi et al., 2 Oct 2025).

Generative inverse sampling regimes therefore require both efficient learning of $p(x)$ (or tractable surrogates) and the design of tailored sampling algorithms to realize draws from $p(x|y)$ under computational constraints and physical priors.

2. Algorithmic Paradigms and Sampling Strategies

A wide spectrum of algorithmic paradigms enables generative inverse sampling, varying mainly in the form and internal structure of the generative prior, and the associated inference/sampling strategy.

• Diffusion Posterior Sampling (DPS): Utilizes reverse-time SDEs (or ODEs) matched to an unconditional diffusion prior $p_0(x)$, modified at each step by a measurement-informed gradient (usually via a “Tweedie’s estimate” for the clean sample, plus a data-consistency gradient), resulting in updates of the form

$$x_{i-1} = x_i + \alpha_i\,s_\theta(x_i,i) + \beta_i\,\nabla_{x_i}\log p(y \mid \hat x_0(x_i)) + \sqrt{\gamma_i}\,z_i,$$

for appropriate schedules, where $s_\theta$ is the score network (Chung et al., 2022, Aali et al., 2024, Meng et al., 15 Jun 2025).

• Flow-based Posterior Sampling: Employs invertible or ODE-driven flows (e.g., “flow matching” or rectified flows) to generate a continuous path from a simple reference to the target data law. Posterior sampling modifies this path with a likelihood gradient or combined Eulerian/Langevin+proximal updates for manifold consistency, balancing mode seeking and exploration (Kim et al., 11 Mar 2025, Park et al., 8 Dec 2025).
• Plug-and-Play (PnP) and Proximal Generators: Uses modular priors and likelihoods realized via “proximal generators,” such as image denoisers for the prior and optimization/analytic maps for the forward model. Alternating updates sample from appropriately regularized conditional distributions (proximal distributions), generalizing classical ADMM to generative sampling (Bouman et al., 2023).
• Latent-space and Monte Carlo Methods: Sampling in the latent space of pre-trained autoencoders or VAEs, employing score-based or SMC/particle filtering methods to compute posterior samples robustly, especially in the presence of nonlinear decoder mappings (Achituve et al., 9 Feb 2025, Daras et al., 2022).
• Consistency Models and Fast Samplers: Employ fast, few-step generators trained for consistency across noise scales; measurement guidance is enforced via stochastic projection or Tikhonov regularization at each step, providing rapid yet effective posterior approximation (Tanevardi et al., 2 Oct 2025).

A summary of several leading approaches is provided below:

Method	Generative Prior	Sampling Principle
DPS/Score-based diffusion	Diffusion (score match)	SDE/ODE + gradient guidance
FlowDPS/FlowLPS	ODE-driven flow/rectified	Flow ODE + likelihood/noise split
PnP/GPnP	Denoiser + analytic	Proximal alternation (Gibbs-like)
LD-SMC/SGILO	Latent diffusion/GAN	SMC/Langevin in latent manifold
Consistency Models	ConsistencyNet/DDIM	Fast projection-guided sampling

3. Ensuring Posterior Consistency and Data Fidelity

Posterior consistency in generative inverse sampling hinges on the interplay between generative manifold preservation and fidelity to observed data. State-of-the-art techniques introduce explicit and implicit mechanisms for enforcing both:

• Triple Consistency (e.g., SITCOM): Enforces data-fitting, network (backward) consistency, and forward-diffusion consistency at each step, leveraging gradient-based optimization to solve

$$\min_v \| \mathcal{A}(f(v; t)) - y \|_2^2 + \lambda \| v - x_t \|_2^2,$$

where $f(v; t)$ is the network denoiser mapping. This enables measurement-consistent, forward-consistent stochastic sampling in few reverse steps (Alkhouri et al., 2024).

• Guidance via Likelihood Gradients: Posterior sampling in high-dimensional nonlinear/noisy settings is achieved by adding the gradient of $\log p(y|x)$ with respect to either the current iterate or the estimated clean sample, avoiding unstable hard projections and maintaining the sample within the generative prior’s manifold (Chung et al., 2022, Alkhouri et al., 2024, Kim et al., 11 Mar 2025).
• Manifold Anchoring and Proximal Steps: For flow-based models, a hybrid of Langevin steps (exploring the measure while adhering to the prior manifold) and proximal/MAP updates (anchoring to high-likelihood regions) maximizes posterior accuracy and sample realism (Park et al., 8 Dec 2025).
• Uncertainty Quantification and Diversity: Approaches such as GUIDe and DAISI compute full posterior distributions—not only point estimates—enabling coverage of feasible designs, uncertainty quantification, and robust out-of-distribution generalization (Andrae et al., 29 Nov 2025, Mu et al., 6 Sep 2025).

4. Representative Applications and Problem Domains

Generative inverse sampling underpins modern advances in a range of scientific and engineering domains, each imposing specific structural or computational requirements:

• Imaging and Vision: Inpainting, denoising, deblurring, super-resolution, compressed sensing MRI (incl. ambient/corrupted training), and tomography—enabling both classic and challenging nonlinear reconstructions (Bouman et al., 2023, Aali et al., 2024, Chung et al., 2022, Meng et al., 15 Jun 2025).
• Scientific Data Assimilation: Filtering in high-dimensional state-space models for geosciences, radar nowcasting, and PDE-governed systems, leveraging generative flows and stochastic interpolants for non-Gaussian, multimodal uncertainty representation (Andrae et al., 29 Nov 2025, Lin et al., 30 Jan 2026).
• Inverse Design and Engineering: Discovering diverse and feasible solutions in mechanical, structural, and materials design, often conditioning on target functional responses (e.g., desired stress-strain curves, fluid properties) via generative MCMC or diffusion-guided Boltzmann sampling (Mu et al., 6 Sep 2025, Vlastelica et al., 2023, Dan et al., 2019).
• Seismic and Physical Sciences: Posterior sampling for noisy, ill-posed geophysical inverse problems (e.g., denoising/interpolating seismic traces), leveraging accelerated diffusion models with power-law noise schedules for rapid, high-fidelity reconstruction (Meng et al., 15 Jun 2025).

5. Theoretical Guarantees and Limitations

Theoretical analyses establish ergodicity, stationary measure correctness, and polynomial mixing under various algorithmic regimes:

• Markov Chain Reversibility: Proximal alternation (e.g., GPnP) yields reversible Markov chains with explicit characterizations of stationary joint/posterior densities (Bouman et al., 2023).
• Stability under Noise and Nonlinearity: Avoiding hard measurement projections improves stability in measurement-noise settings; manifold-guided gradients maintain posterior accuracy (Chung et al., 2022).
• Unbiased Sampling: Approaches relying on SMC and Langevin MCMC can, in the limit, achieve asymptotic exactness in their sample approximation to the true Bayesian posterior (Park et al., 8 Dec 2025, Achituve et al., 9 Feb 2025).
• Data Efficiency: Decoupled (prior/forward) models (e.g., DDIS) maintain accuracy under extremely sparse or unpaired data, avoiding the degeneration of joint-embedded schemes (Lin et al., 30 Jan 2026).

Limitations include:

• Task- and regime-specific hyperparameter sensitivity (e.g., annealing, noise level selection, regularization),
• Potential for local minima entrapment in highly nonlinear or underdetermined settings (especially with per-step optimization),
• Computational cost in high-dimensional domains, especially for SMC-based samplers,
• Model mismatch or prior misspecification risks, particularly for out-of-distribution targets,
• Need for forward operator knowledge (non-blind setting) and difficulties in joint estimation in blind scenarios (Alkhouri et al., 2024, Andrae et al., 29 Nov 2025).

6. Recent Advances and Future Directions

Active areas of research and anticipated advances include:

• Latent-space and Transformer-based Flows: FlowDPS and similar frameworks operate directly in high-dimensional latent spaces (e.g., Stable Diffusion 3.0 latent backbone), achieving scalable, high-fidelity, zero-shot posterior inference without retraining (Kim et al., 11 Mar 2025, Park et al., 8 Dec 2025).
• Accelerated and Adaptive Sampling: Power-law noise schedules, DDIM-style non-Markov updates, and learned step-size adaptation enable order-of-magnitude reductions in function evaluations for both unconditional and posterior tasks (Meng et al., 15 Jun 2025, Aali et al., 2024, Alkhouri et al., 2024).
• Cross-corruption and Domain Adaptation: “Ambient” priors, trained on corrupted data, outperform clean-trained models in highly corrupted regimes (e.g., undersampled MRI); this regime-dependent “prior crossover” suggests further work in meta- and transfer-prior learning (Aali et al., 2024).
• Theoretical Analysis of Guidance and Score Approximation: Addressing theoretical gaps in surrogates for intractable likelihood gradients, especially in nonlinear/noisy posteriors, remains pivotal for rigorously characterizing sampler bias (Chung et al., 2022, Kim et al., 11 Mar 2025).
• Multimodal and OOD Extension: Ensuring algorithmic coverage of all feasible solution modes, handling OOD targets, and integrating higher-order solvers and ensemble/tempering strategies are explicit future research aims (Alkhouri et al., 2024, Mu et al., 6 Sep 2025, Andrae et al., 29 Nov 2025).

These directions aim to further unify generative Bayesian inference with modular, physics-agnostic architectures, ultimately providing computationally tractable, physically consistent, uncertainty-quantifying solvers for inverse problems across the sciences and engineering.