Zhaoqiang Liu, Tongyao Pang, Ruibing Wang +1stat.ML cs.AI cs.LG
Pretrained diffusion models represent image distributions through a continuum of progressively smoothed distributions. This multiscale structure organizes generation from global structure to fine detail and supports high-quality, diverse samples. We exploit the same multiscale diffusion prior for linear imaging inverse problems. Rather than using the pretrained model only as a denoiser in an outer iteration, we define a surrogate likelihood whose center is aligned with the clean-image coordinate and whose covariance accounts for residual diffusion uncertainty. This construction defines an explicit surrogate posterior path, from which we derive continuous posterior dynamics. A tunable Langevin component supports target tracking and allows the amount of posterior exploration to be adapted to the application. We prove endpoint consistency and a finite-horizon tracking bound and, in the exact-score setting, first-order weak accuracy. For computation, we derive the Posterior-Dynamics Implicit--Explicit sampler (PD-IMEX), a stable method using one score evaluation per diffusion scale and an implicit data-consistency update. Experiments on deblurring, super-resolution, and inpainting show strong reconstruction quality at 100 score evaluations, coarse-grid stability, and controllable fidelity--diversity behavior.
Latent Flow Models have revolutionized compressed-space image synthesis, yet their application to high-fidelity inverse problems remains bottlenecked. In this paper, we trace this dilemma to a fundamental geometric limitation of pre-trained autoencoders, which we term \emph{First-Order Manifold Blindness}. Severe decoder compression (e.g., retaining only $\sim\!2\%$ of the original degrees of freedom) produces a rank-deficient Jacobian, rendering high-frequency measurement residuals in its orthogonal complement invisible to latent gradients even when the decoder can represent the target image. To overcome this bottleneck, we propose Hybrid-Domain Posterior Sampling (HDPS), a decoupled inference framework that disentangles physical measurement consistency from semantic prior modeling. HDPS diverges into the pixel space, leveraging Langevin dynamics to absorb precise orthogonal measurement gradients, and subsequently projects these structural corrections back onto the generative manifold. An optimization-based latent alignment is introduced to filter pixel-space artifacts while avoiding the semantic drift of direct encoding. Extensive experiments on diverse inverse problems demonstrate that HDPS establishes a new state-of-the-art, successfully recovering the high-frequency structural precision that latent-only solvers inherently discard. The code is available at \href{https://github.com/74587887/HDPS}{https://github.com/74587887/HDPS}.
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.
Yuchen Jiao, Na Li, Changxiao Cai +2cs.LG cs.AI stat.ML
Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.
Federico Carrara, Aman Kukde, Melisande Croft +2cs.CV cs.AI
SWITi is a test-time method for reducing artifacts in tiled predictions, particularly for neural networks that learn posterior distributions from which solutions are sampled at inference time. Tiled predictions are unavoidable for large image data, and artifacts arise whenever tiles are smaller than a network's receptive field and when tiles are independent posterior samples. SWITi averages overlapping sliding-window predictions, so discrepancies between neighboring samples are spread across shifted tile positions rather than accumulating at fixed seam coordinates. For posterior models, SWITi uses no more tile samples than an MMSE estimate requires and therefore incurs no additional forward passes. Additionally, we introduce two reference-free metrics, the Fraction of Rejected Tests (FRT) and Artifact Severity (ASV), for detecting and quantifying tiling artifacts from a per-tile permutation test that compares the distribution of pixel gradients across tile seams against the surrounding image content. On pre-trained and published image splitting models across three fluorescence microscopy datasets in 2D and 3D, we show that SWITi substantially attenuates stitching seams while also improving reconstruction fidelity and resolution. Since tiling artifacts in posterior predictions can easily be mistaken for biological structures or for boundaries between biological structures, removing or reducing them using SWITi will improve the downstream processing of large image predictions, which is particularly relevant for biomedical data.
Magnetic resonance imaging comes in various modality contrasts that provide complementary anatomical and pathological information. Complete multimodal acquisitions are often unavailable due to time and protocol constraints. This leads to real-world datasets with missing modalities, where conventional medical image translation methods are typically limited to fixed source-target settings or require retraining for each observed source-target pair. We propose a unified framework that formulates missing-modality generation as a linear inverse problem under a joint distribution and solves it via posterior sampling with a flow matching model. By learning a joint prior over the complete modality set, our method can reconstruct arbitrary missing modalities at inference time by guiding the sampling trajectory to enforce measurement consistency with observed modalities. We further mitigate inter-modality error propagation in multi-target generation by adopting a many-to-one sampling strategy. Experiments on BraTS and IXI datasets show that our method achieves the best performance over baselines across most missing-modality scenarios. In downstream tumor segmentation, synthesized images from our method result in higher segmentation performance, indicating better preservation of clinically relevant structures.
A growing family of training-free solvers -- FlowDPS, FLOWER, PnP-Flow and their diffusion ancestors (DPS, DAPS) -- repurpose a pretrained flow-matching prior to solve imaging inverse problems by adding a measurement-guidance term to the deterministic probability-flow ODE. Despite strong empirical results, what these per-step corrections actually approximate -- and how far the resulting samples are from the true posterior $p(x\mid y)$ -- has not been characterized. We give a posterior-transport account of flow-based inverse problem solving. Our starting point is a simple but consequential fact: for a \emph{deterministic} flow prior, Bayesian conditioning is realized entirely by a \emph{reweighting of the source distribution}, not by a drift correction; pushing the reweighted source through the \emph{unmodified} velocity field yields exact posterior samples. From this we show that trajectory-guidance solvers can be read as the minimum-kinetic-energy \emph{correction} field needed to morph the unconditional source into the posterior, and that FlowDPS / FLOWER / PnP-Flow correspond to distinct zeroth-order / Gaussian / proximal approximations of this single object; we bound the resulting posterior bias in Wasserstein distance. A controlled $2$D study with a closed-form posterior confirms the theory decisively: source reweighting matches the true posterior to the Monte-Carlo floor on every metric, whereas trajectory guidance incurs $200$--$800\times$ larger error and collapses posterior modes, \emph{regardless of guidance strength}. Guided by the analysis we propose a cheap, principled velocity-correction solver that is competitive across two in-domain priors (AFHQ, CelebA) and two out-of-distribution settings while, unlike point-estimate source-space optimizers, producing diverse posterior samples with uncertainty that correlates with reconstruction error.
Yaozhong Shi, Zachary E. Ross, Yisong Yuestat.ML cs.AI cs.LG
Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FAPS produces coherent posterior samples with accurate uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better PDE noisy inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.
Variational inference (VI) is a powerful method for principled posterior inference for scientific inverse imaging. VI learns the posterior distribution, often with a flow-based network, which can cheaply generate posterior samples upon optimization, and can flexibly incorporate score-based or classic priors. However, its application to large-scale image reconstruction is severely hindered by the poor scalability of the flow-based networks. In this work, we introduce ShuffleFlow, a scalable VI framework to address this challenge. Our method breaks down the problem into three parts: a pixel-unshuffling-based image coordinate sampler, a neural field as feature encoder, and a conditional normalizing flow (CNF) as posterior estimator. Specifically, our framework partitions an image into a stack of sub-images with pixel-unshuffling and uses a shared CNF to model the joint distribution of the sub-image stack. We condition the CNF on the output of a neural field, which embeds feature vectors corresponding to pixel-unshuffling sample locations to capture spatial structures, and share the flow's latent variable across the channels to model their correlations. We demonstrate our method's effectiveness and efficiency on both linear and nonlinear imaging inverse problems, and show its ability to more rapidly generate a high-sample-count posterior than diffusion samplers.
We study the privacy of releasing posterior sample paths from a Gaussian process (GP) when the entire training set including covariates and responses is private. Unlike standard differential-privacy (DP) mechanisms that add external noise, posterior sampling is random by construction. We show that this intrinsic randomness yields DP guarantees by deriving explicit Rényi-DP bounds for GP posterior sample-path release. The bounds separate posterior-mean leakage from data-dependent posterior-covariance leakage showing that meaningful privacy depends sharply on effective ridge regularisation. We apply membership-inference attacks to show that empirical leakage follows the predicted dependence on regularisation, posterior variance and the number of released posterior sample-paths. Utility experiments on downstream posterior-sampling tasks identify noisy-observation regimes where privacy-compatible regularisation preserves useful decisions with modest utility loss. When stronger privacy is needed, the intrinsic guarantee can be sharpened by adding calibrated GP noise, providing an explicit additional privacy knob.
Abbas Mammadov, Ozgur Kara, Kaan Oktay +5cs.LG cs.CV stat.ML
Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.
Diffusion models provide expressive data-driven priors for Bayesian inverse problems, but many diffusion posterior samplers rely on heuristic guidance approximations that can fail for nonlinear operators and multimodal posteriors. In this work, we develop a stabilized path-space framework for diffusion-based posterior sampling. Starting from a base diffusion process whose terminal marginal represents the prior, we define a likelihood-weighted target measure on trajectories and cast posterior sampling as learning a controlled stochastic process whose path measure matches this target. This formulation connects diffusion posterior sampling to stochastic optimal control while preserving the Bayesian structure needed for uncertainty quantification. We introduce a time reparameterization that makes the path-space control problem well posed by removing the bias induced by the unknown initial value function, without auxiliary training. We then learn the control via a trust-region path-space optimization method with log-variance objectives. The path-space perspective also unifies our learned control approach with existing guidance-based samplers, quantifies the sampling error induced by approximate controls, and yields importance sampling corrections for asymptotically exact posterior expectations. We evaluate the proposed framework on a suite of benchmark inverse problems with analytically characterized or high-quality reference posteriors, enabling principled assessment of sampling accuracy and uncertainty quantification. These experiments provide insight into the behavior of diffusion-based posterior samplers and demonstrate improved accuracy and robustness over leading approaches.
Zhengkai Pan, Peter Potaptchik, Wenxi Yao +2stat.ML cs.LG
Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.
We study model-free Q-learning in finite-horizon episodic Markov Decision Processes (MDPs) with stationary dynamics across episodes. We identify a central issue in nascent model-free posterior-sampling works: the reliance on delayed learning in order to prove theoretical guarantees. In particular, we identify three opportunities for faster learning - (i) value-function update order, (ii) update frequencies, and (iii) value-function initialization. Using Wang et al.'s RandomizedQ as a basis, we illustrate these changes and their individual (as well as cumulative) impact in multiple empirical studies. We find that our combined modifications, termed ReversedQ, improve scaled mean cumulative reward compared to RandomizedQ, from 9.53% to 78.78% in the Bidirectional Diabolical Combination Lock (BDCL), and from 21.76% to 61.81% in a chain MDP.