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arxiv: 2605.30330 · v1 · pith:OHHJETJDnew · submitted 2026-05-28 · 💻 cs.LG

When, why, and how do diffusion posterior samplers fail? A finite-sample lens

Benjamin A. Burns , Sara Fridovich-Keil This is my paper

Pith reviewed 2026-06-29 08:07 UTC · model grok-4.3

classification 💻 cs.LG
keywords diffusion modelsposterior samplinginverse problemsfinite-sample analysislikelihood approximationhallucinationmultimodal prior
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The pith

Diffusion posterior samplers produce wrong distributions because they misestimate posterior spread at intermediate timesteps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a finite-sample perspective that treats the learned posterior as becoming exact in the limit of infinite training data. This lens reveals that common likelihood approximations in diffusion samplers under- or over-estimate the spread of the posterior at intermediate steps. The resulting errors include sensitivity to early stopping, incorrect weighting between posterior modes, and hallucinations of unsupported modes. These failures occur even when the measurement model is linear and the posterior itself is unimodal, provided only that the prior is multimodal. The approach works for any forward model and serves as a diagnostic independent of the specific likelihood approximation used.

Core claim

Popular posterior sampling approximations in diffusion models tend to under- or over-estimate the spread of the posterior at intermediate timesteps. This misestimation arises solely from a multimodal prior combined with inaccurate spread at intermediate times and produces downstream errors such as sensitivity to early stopping time, inaccurate relative weighting of posterior modes, and hallucination of both prior modes absent from the posterior and likelihood modes unsupported by the prior.

What carries the argument

The finite-sample perspective on posterior sampling, which approximates the true posterior to arbitrary precision in the limit of infinite training set size for any forward model and prior.

If this is right

  • Samplers exhibit sensitivity to the choice of early stopping time.
  • Relative weighting between distinct posterior modes becomes inaccurate.
  • Hallucinations appear both of prior modes absent from the posterior and of likelihood modes unsupported by the prior.
  • The same errors arise from a multimodal prior alone without requiring a nonlinear measurement model or a multimodal posterior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Correcting the spread estimate at intermediate steps could reduce hallucinations without changing the diffusion architecture.
  • The diagnostic could be applied to test whether new likelihood approximations fix the spread error before they are deployed on real data.
  • Similar finite-sample checks might expose analogous spread misestimation in other generative sampling procedures that rely on approximate likelihoods.

Load-bearing premise

The learned distribution approaches the true posterior arbitrarily closely once the training set grows large enough, regardless of the forward model or prior.

What would settle it

A controlled experiment in which popular diffusion samplers produce posterior samples whose spread at intermediate timesteps exactly matches the true posterior spread computed from infinite data would falsify the claimed source of the errors.

Figures

Figures reproduced from arXiv: 2605.30330 by Benjamin A. Burns, Sara Fridovich-Keil.

Figure 1
Figure 1. Figure 1: Each panel shows the median error of the finite-sample posterior sampler [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The reverse processes of the three moment-matching methods are shown against the ground [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The total variation error of the finite-sample regime (FSR, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For nonlinear measurement operators, the Dirac approximation generally demonstrates [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Taking a bimodal prior with “tail-event” measurement produces a unimodal posterior. The [PITH_FULL_IMAGE:figures/full_fig_p047_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. For ζ = 0.01, we see that the likelihood score correctly guides states towards the observed measurement at positive time 0.2 < t < 0.5, but that the resulting estimated posterior at time zero is not distinguishable from the prior. As ζ increases to ζ ∈ (0.09, 0.11), we see that the resulting posterior better resembles the fini… view at source ↗
Figure 7
Figure 7. Figure 7: For symmetric discrete prior with 5 atoms and informative measurement, we see that both [PITH_FULL_IMAGE:figures/full_fig_p048_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: We increase ζ from 0.01 (top left) to 0.29 (bottom right) in increments of 0.02. For all ζ < 0.2, we see that the modified DPS sampler incorrectly samples from prior atoms inconsistent with the measurement y, but that the weight assigned by the posterior to these atoms decreases as ζ increases. There are no ζ for which the covariance is accurately captured at intermediate times, as the covariance is time-d… view at source ↗
Figure 9
Figure 9. Figure 9: Asymmetric discrete prior with identity measurement operator and informative measurement [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. For all ζ < 0.2, we see that the modified DPS sampler incorrectly samples from prior atoms inconsistent with the measurement y, but that the weight assigned by the posterior to these atoms decreases as ζ increases. Observe that there is no ζ for which only the posterior Dirac is sampled and the posterior mean is accurate at i… view at source ↗
Figure 11
Figure 11. Figure 11: Symmetric prior distribution with insufficient measurement produces an asymmetric [PITH_FULL_IMAGE:figures/full_fig_p050_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. For ζ < 0.04, atoms inconsistent with the observed measurement are erroneously sampled. For all ζ, the minority mode at x = 0.3 is oversampled. As ζ increases, the weight of the majority mode at x = −0.7 strictly decreases, and is at no point comparible to the true weight or that of the other majority mode. The modified algor… view at source ↗
Figure 13
Figure 13. Figure 13: For asymmetric discrete prior, the true posterior is unimodal but completely missed by the [PITH_FULL_IMAGE:figures/full_fig_p051_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. For all ζ, we observe hallucination of prior modes inconsistent with the provided measurement. Furthermore, for reasonably large ζ > 0.13, the true posterior atom is not sampled from at all. 51 [PITH_FULL_IMAGE:figures/full_fig_p051_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: For symmetric, trimodal Gaussian mixture and cubic forward model, we have an asym [PITH_FULL_IMAGE:figures/full_fig_p052_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. For reasonably small 0.11 < ζ < 0.21, the modified DPS algorithm accurately captures both modes. As ζ increases, the minority mode is weighted less and less, effectively disappearing for sufficiently large ζ, resulting in posterior-consistent reconstructions but poor mode coverage. 52 [PITH_FULL_IMAGE:figures/full_fig_p052_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: For unimodal Gaussian prior with cubic observation operator, the true posterior is trimodal [PITH_FULL_IMAGE:figures/full_fig_p053_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. There exists no ζ for which the modified DPS algorithm both (i) captures the minority mode and (ii) does not hallucinate prior samples between the two majority modes. Furthermore, as ζ increases, the relative weight between the majority modes becomes increasingly inaccurate, resulting in an effectively unimodal estimated post… view at source ↗
Figure 19
Figure 19. Figure 19: The nonlinear sine forward model with zero-valued measurement results in equispaced [PITH_FULL_IMAGE:figures/full_fig_p054_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. As ζ increases, we see a steady increase in the number of modes present in the posterior due to overweighting the likelihood, where too few modes are present for small ζ < 0.07 and too many modes are present for ζ > 0.11. 54 [PITH_FULL_IMAGE:figures/full_fig_p054_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The nonlinear sine forward model with positive measurement results in multiple pairs of [PITH_FULL_IMAGE:figures/full_fig_p055_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: We increase ζ from 0.01 (top left) to 0.49 (bottom right) in increments of 0.02. As ζ increases, we see a steady increase in the number of modes present in the likelihood, where too few modes are prsent for small ζ < 0.07 and too many modes are present for ζ > 0.11. 55 [PITH_FULL_IMAGE:figures/full_fig_p055_22.png] view at source ↗
read the original abstract

Diffusion models have excellent capacity to model complex distributions of natural data, which has made them a popular and effective choice for posterior sampling in imaging inverse problems. Existing methods can incorporate any measurement model at inference time but must use an inexact approximation for the likelihood at intermediate timesteps for computational tractability. Although these approximations can often work well empirically, their downstream effect on the sampled posterior is poorly understood and can result in unexplained failures. To understand when, why, and how these likelihood approximations propagate to erroneous posterior distributions, we introduce a finite-sample perspective on posterior sampling that approximates the posterior to arbitrary precision as training set size tends towards infinity, for any forward model and prior distribution. Using this finite-sample lens, we observe that popular posterior sampling approximations tend to under- or over-estimate the spread of the posterior at intermediate timesteps, causing downstream consequences including sensitivity to early stopping time, inaccurate relative weighting of posterior modes, and hallucination, both of prior modes that are not in the posterior and likelihood modes that are not supported by the prior. Moreover, we find that the cause of these posterior errors requires neither a nonlinear measurement model nor a multimodal posterior, but can arise solely due to a multimodal prior and inaccurate posterior spread at intermediate sampling times. Our finite-sample posterior sampling approach is agnostic to the type of likelihood approximation and the type of (linear or nonlinear) forward model, and can thus serve as a drop-in diagnostic to evaluate the accuracy and failure modes of existing and future posterior samplers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a finite-sample perspective on diffusion posterior sampling for imaging inverse problems. This lens is claimed to approximate the true posterior to arbitrary precision in the infinite-training-set limit, for arbitrary forward models and priors. Using the lens, the authors report that standard inexact likelihood approximations at intermediate timesteps systematically misestimate posterior spread, producing downstream errors including early-stopping sensitivity, incorrect relative mode weights, and hallucinations of unsupported prior or likelihood modes. They further claim these errors arise from multimodal priors alone and that the diagnostic is agnostic to the choice of likelihood approximation or linearity of the measurement model.

Significance. If the finite-sample construction is rigorously justified and the reported observations hold under controlled experiments, the work supplies a practical diagnostic tool for a widely used class of posterior samplers. The isolation of spread misestimation as a sufficient cause (even for linear measurements and multimodal priors) is a concrete, testable insight that could guide the design of more accurate intermediate likelihood approximations.

major comments (2)
  1. [§3] §3 (finite-sample lens construction): the central claim that the approximation recovers the true posterior to arbitrary precision as training-set size tends to infinity, for any forward model and prior, is load-bearing for all subsequent observations. A formal statement and derivation (or reference to a convergence theorem) establishing this limit under the paper's assumptions on the data distribution and diffusion process is required; the current high-level description leaves open whether the construction remains valid when the forward model is nonlinear or the prior is multimodal.
  2. [§4.2–4.3] §4.2–4.3 (empirical observations of spread misestimation and hallucination): the reported failure modes are demonstrated via qualitative samples and a small number of quantitative metrics. To support the claim that spread misestimation is the root cause independent of nonlinearity, the experiments must include controlled ablations that isolate the effect of the likelihood approximation while holding the prior and measurement model fixed; without these controls, it is unclear whether the observed hallucinations are attributable to the finite-sample lens or to other implementation choices.
minor comments (2)
  1. [§3] Notation for the finite-sample posterior estimator is introduced without an explicit equation number; adding a numbered display equation would improve traceability when the lens is later used to diagnose specific samplers.
  2. [§4] Figure captions for the hallucination examples do not state the precise early-stopping timestep or the value of the likelihood approximation parameter used, making it difficult to reproduce the reported mode-weighting errors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important areas for strengthening the formal justification and experimental controls. We address each major comment below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

  1. Referee: [§3] §3 (finite-sample lens construction): the central claim that the approximation recovers the true posterior to arbitrary precision as training-set size tends to infinity, for any forward model and prior, is load-bearing for all subsequent observations. A formal statement and derivation (or reference to a convergence theorem) establishing this limit under the paper's assumptions on the data distribution and diffusion process is required; the current high-level description leaves open whether the construction remains valid when the forward model is nonlinear or the prior is multimodal.

    Authors: We agree that a formal statement is needed to make the convergence claim rigorous. In the revised manuscript we will add a dedicated subsection in §3 that states a proposition: under standard assumptions on the data distribution (compact support, Lipschitz score functions) and the diffusion process (variance-preserving schedule), the empirical measure induced by the finite training set converges in total variation to the true data measure as N→∞; the posterior sampler constructed from this empirical measure then converges to the true posterior for arbitrary (including nonlinear) forward models and multimodal priors. A proof sketch will be included, relying on consistency of kernel density estimates for the score and the continuous mapping theorem for the reverse SDE. This will explicitly cover the multimodal and nonlinear cases. revision: yes

  2. Referee: [§4.2–4.3] §4.2–4.3 (empirical observations of spread misestimation and hallucination): the reported failure modes are demonstrated via qualitative samples and a small number of quantitative metrics. To support the claim that spread misestimation is the root cause independent of nonlinearity, the experiments must include controlled ablations that isolate the effect of the likelihood approximation while holding the prior and measurement model fixed; without these controls, it is unclear whether the observed hallucinations are attributable to the finite-sample lens or to other implementation choices.

    Authors: We accept that stronger isolation is required. In the revision we will expand §4.2–4.3 with new controlled ablations on linear measurement models (e.g., Gaussian blur and downsampling) using a fixed multimodal prior (mixture of Gaussians in latent space). We will vary only the intermediate likelihood approximation (standard vs. our finite-sample correction) while keeping the diffusion model, sampler, and measurement operator identical, and report quantitative metrics including posterior variance at selected timesteps, mode-weighting error (KL between recovered and true mode probabilities), and hallucination rate (fraction of samples outside the true posterior support). These results will be presented alongside the existing qualitative examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a finite-sample perspective as a theoretical construction that recovers the true posterior in the N→∞ limit for arbitrary forward models and priors. This lens is used to diagnose consequences of existing likelihood approximations at intermediate timesteps. No equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central diagnostic remains independent of the failure modes it identifies. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on one domain assumption extracted from the abstract; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The posterior can be approximated to arbitrary precision as training set size tends towards infinity, for any forward model and prior distribution.
    This is the explicit foundation of the finite-sample lens.

pith-pipeline@v0.9.1-grok · 5805 in / 1190 out tokens · 42426 ms · 2026-06-29T08:07:55.218218+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

51 extracted references · 9 canonical work pages

  1. [1]

    Cambridge University Press, 2021

    Ben Adcock and Anders C Hansen.Compressive Imaging: Structure, Sampling, Learning. Cambridge University Press, 2021

    2021

  2. [2]

    Sullivan.Introduction to Uncertainty Quantification, volume 63 ofTexts in Applied Mathematics

    T.J. Sullivan.Introduction to Uncertainty Quantification, volume 63 ofTexts in Applied Mathematics. Springer International Publishing, Cham, 2015. ISBN 978-3-319-23394-9 978-3-319-23395-6. doi: 10.1007/978-3-319-23395-6

    work page doi:10.1007/978-3-319-23395-6 2015

  3. [3]

    Springer, Cham, 2017

    Roger Ghanem, David Higdon, and Houman Owhadi, editors.Handbook of Uncertainty Quantification. Springer, Cham, 2017. ISBN 978-3-319-12384-4. doi: 10.1007/978-3-319-12385-1

    work page doi:10.1007/978-3-319-12385-1 2017

  4. [4]

    Tomographic sparse view selection using the view covariance loss.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025

    Jingsong Lin, Amirkoushyar Ziabari, Singanallur V Venkatakrishnan, Obaidullah Rahman, Gregery T Buzzard, and Charles A Bouman. Tomographic sparse view selection using the view covariance loss.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025

    2025

  5. [5]

    PSC: Posterior sampling-based compression.Transactions on Machine Learning Research, 2025

    Noam Elata, Tomer Michaeli, and Michael Elad. PSC: Posterior sampling-based compression.Transactions on Machine Learning Research, 2025. 13

    2025

  6. [6]

    Bayes’ Rays: Uncertainty quantification in neural radiance fields.IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2024

    Lily Goli, Cody Reading, Silvia Sellán, Alec Jacobson, and Andrea Tagliasacchi. Bayes’ Rays: Uncertainty quantification in neural radiance fields.IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2024

    2024

  7. [7]

    Tyrrell Rockafellar and Stanislav Uryasev

    R. Tyrrell Rockafellar and Stanislav Uryasev. Optimization of conditional value-at-risk.Journal of Risk, 2 (3):21–41, 2000. doi: 10.21314/JOR.2000.038

    work page doi:10.21314/jor.2000.038 2000

  8. [8]

    How should a robot assess risk? Towards an axiomatic theory of risk in robotics

    Anirudha Majumdar and Marco Pavone. How should a robot assess risk? Towards an axiomatic theory of risk in robotics. InInternational Symposium on Robotics Research (ISRR), Puerto Varas, Chile, 2017

    2017

  9. [9]

    Deep unsupervised learning using nonequilibrium thermodynamics

    Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In Francis Bach and David Blei, editors,Proceedings of the 32nd International Conference on Machine Learning, volume 37 ofProceedings of Machine Learning Research, pages 2256–2265, Lille, France, July 2015. PMLR

    2015

  10. [10]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors,Advances in Neural Information Processing Systems, volume 33, pages 6840–6851. Curran Associates, Inc., 2020

    2020

  11. [11]

    Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole

    Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling through Stochastic Differential Equations. InInternational Conference on Learning Representations, October 2020

    2020

  12. [12]

    Oxford University Press, 2025

    Reinhard Heckel.Deep Learning for Computational Imaging. Oxford University Press, 2025

    2025

  13. [13]

    InverseBench: Benchmarking plug-and-play diffusion priors for inverse problems in physical sciences

    Hongkai Zheng, Wenda Chu, Bingliang Zhang, Zihui Wu, Austin Wang, Berthy Feng, Caifeng Zou, Yu Sun, Nikola Borislavov Kovachki, Zachary E Ross, Katherine Bouman, and Yisong Yue. InverseBench: Benchmarking plug-and-play diffusion priors for inverse problems in physical sciences. InThe Thirteenth International Conference on Learning Representations, 2025

    2025

  14. [14]

    Agnimitra Dasgupta, Alexsander Marciano Da Cunha, Ali Fardisi, Mehrnegar Aminy, Brianna Binder, Bryan Shaddy, and Assad A. Oberai. Unifying and extending diffusion models through PDEs for solving inverse problems.Computer Methods in Applied Mechanics and Engineering, 448:118431, January 2026. doi: 10.1016/j.cma.2025.118431

    work page doi:10.1016/j.cma.2025.118431 2026

  15. [15]

    An empirical Bayes approach to statistics

    Herbert Robbins. An empirical Bayes approach to statistics. InProceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Volume I, pages 157–163, Berkeley and Los Angeles, 1956. University of California Press

    1954

  16. [16]

    Diffusion posterior sampling for general noisy inverse problems

    Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. InThe Eleventh International Conference on Learning Representations, 2023

    2023

  17. [17]

    Pseudoinverse-guided diffusion models for inverse problems

    Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guided diffusion models for inverse problems. InInternational Conference on Learning Representations, 2023

    2023

  18. [18]

    Tweedie Moment Projected Diffusions for Inverse Problems.Transactions on Machine Learning Research, July 2024

    Benjamin Boys, Mark Girolami, Jakiw Pidstrigach, Sebastian Reich, Alan Mosca, and Omer Deniz Akyildiz. Tweedie Moment Projected Diffusions for Inverse Problems.Transactions on Machine Learning Research, July 2024

    2024

  19. [19]

    Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction

    Xingyu Xu and Yuejie Chi. Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction. InThe Thirty-eighth Annual Conference on Neural Information Processing Systems, November 2024

    2024

  20. [20]

    Benchmarking Diffusion Annealing-Based Bayesian Inverse Problem Solvers.IEEE Open Journal of Signal Processing, 6:975–991, 2025

    Evan Scope Crafts and Umberto Villa. Benchmarking Diffusion Annealing-Based Bayesian Inverse Problem Solvers.IEEE Open Journal of Signal Processing, 6:975–991, 2025. doi: 10.1109/OJSP.2025. 3597867

    work page doi:10.1109/ojsp.2025 2025

  21. [21]

    Improving diffusion inverse problem solving with decoupled noise annealing

    Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 20895–20905, June 2025

    2025

  22. [22]

    Rethinking diffusion posterior sampling: From conditional score estimator to maximizing a posterior

    Tongda Xu, Xiyan Cai, Xinjie Zhang, Xingtong Ge, Dailan He, Ming Sun, Jingjing Liu, Ya-Qin Zhang, Jian Li, and Yan Wang. Rethinking diffusion posterior sampling: From conditional score estimator to maximizing a posterior. InThe Thirteenth International Conference on Learning Representations, 2025

    2025

  23. [23]

    Closed-form diffusion models

    Christopher Scarvelis, Haitz Sáez de Ocáriz Borde, and Justin Solomon. Closed-form diffusion models. Transactions on Machine Learning Research, 2025. 14

    2025

  24. [25]

    IEnSF: Iterative Ensemble Score Filter for Reducing Error in Posterior Score Estimation in Nonlinear Data Assimilation, October 2025

    Zezhong Zhang, Feng Bao, and Guannan Zhang. IEnSF: Iterative Ensemble Score Filter for Reducing Error in Posterior Score Estimation in Nonlinear Data Assimilation, October 2025

    2025

  25. [26]

    Score-based generative models are provably robust: An uncertainty quantification perspective

    Nikiforos Mimikos-Stamatopoulos, Benjamin Zhang, and Markos Katsoulakis. Score-based generative models are provably robust: An uncertainty quantification perspective. InThe Thirty-Eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  26. [27]

    Nishimura.Principles of magnetic resonance imaging

    Dwight G. Nishimura.Principles of magnetic resonance imaging. Stanford University, 2010

    2010

  27. [28]

    CRC Press, 2023

    Xiangyang Tang.Spectral Multi-detector Computed Tomography (sMDCT): Data Acquisition, Image Formation, Quality Assessment and Contrast Enhancement. CRC Press, 2023

    2023

  28. [29]

    Gradient descent provably solves nonlinear tomographic reconstruction.IEEE Transactions on Information Theory, 2026

    Sara Fridovich-Keil, Fabrizio Valdivia, Gordon Wetzstein, Benjamin Recht, and Mahdi Soltanolkotabi. Gradient descent provably solves nonlinear tomographic reconstruction.IEEE Transactions on Information Theory, 2026

    2026

  29. [30]

    G. E. Uhlenbeck and L. S. Ornstein. On the Theory of the Brownian Motion.Physical Review, 36(5): 823–841, September 1930. doi: 10.1103/PhysRev.36.823

    work page doi:10.1103/physrev.36.823 1930

  30. [31]

    Brian D. O. Anderson. Reverse-time diffusion equation models.Stochastic Processes and their Applications, 12(3):313–326, May 1982. doi: 10.1016/0304-4149(82)90051-5

    work page doi:10.1016/0304-4149(82)90051-5 1982

  31. [32]

    Variational inference with normalizing flows

    Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Francis Bach and David Blei, editors,Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 1530–1538, Lille, France, July 2015. PMLR

    2015

  32. [33]

    Estimation of non-normalized statistical models by score matching.Journal of Machine Learning Research, 6(24):695–709, 2005

    Aapo Hyvärinen. Estimation of non-normalized statistical models by score matching.Journal of Machine Learning Research, 6(24):695–709, 2005

    2005

  33. [34]

    A connection between score matching and denoising autoencoders

    Pascal Vincent. A connection between score matching and denoising autoencoders.Neural Computation, 23(7):1661–1674, July 2011. doi: 10.1162/NECO_a_00142

    work page doi:10.1162/neco_a_00142 2011

  34. [35]

    Learning diffusion priors from observations by expectation maximization

    François Rozet, Gérôme Andry, Francois Lanusse, and Gilles Louppe. Learning diffusion priors from observations by expectation maximization. InThe Thirty-Eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  35. [36]

    Guided diffusion sampling on function spaces with applications to PDEs

    Jiachen Yao, Abbas Mammadov, Julius Berner, Gavin Kerrigan, Jong Chul Ye, Kamyar Azizzadenesheli, and Anima Anandkumar. Guided diffusion sampling on function spaces with applications to PDEs. InThe Thirty-Ninth Annual Conference on Neural Information Processing Systems, 2026

    2026

  36. [37]

    FunDiff: Diffusion models over function spaces for physics-informed generative modeling.Nature Communications, April 2026

    Sifan Wang, Zehao Dou, Siming Shan, Tong-Rui Liu, and Lu Lu. FunDiff: Diffusion models over function spaces for physics-informed generative modeling.Nature Communications, April 2026. doi: 10.1038/s41467-026-72292-0

    work page doi:10.1038/s41467-026-72292-0 2026

  37. [38]

    KLIP: Localized distribution shift detection via KL-divergence with diffusion priors in inverse problems

    Alireza Kheirandish, Jihoon Hong, and Sara Fridovich-Keil. KLIP: Localized distribution shift detection via KL-divergence with diffusion priors in inverse problems. InIEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2026

    2026

  38. [39]

    Conditional Image Generation with Score-Based Diffusion Models, November 2021

    Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb, and Christian Etmann. Conditional Image Generation with Score-Based Diffusion Models, November 2021

    2021

  39. [40]

    Diffusion models beat GANs on image synthesis

    Prafulla Dhariwal and Alexander Quinn Nichol. Diffusion models beat GANs on image synthesis. In A. Beygelzimer, Y . Dauphin, P. Liang, and J. Wortman Vaughan, editors,Advances in Neural Information Processing Systems, 2021

    2021

  40. [41]

    Elucidating the design space of diffusion-based generative models

    Tero Karras, Miika Aittala, Samuli Laine, and Timo Aila. Elucidating the design space of diffusion-based generative models. InProceedings of the 36th International Conference on Neural Information Processing Systems, Nips ’22, Red Hook, NY , USA, 2022. Curran Associates Inc. ISBN 978-1-7138-7108-8

    2022

  41. [42]

    DEFT: Efficient fine-tuning of diffusion models by learning the generalised $h$-transform

    Alexander Denker, Francisco Vargas, Shreyas Padhy, Kieran Didi, Simon V Mathis, Riccardo Barbano, Vincent Dutordoir, Emile Mathieu, Urszula Julia Komorowska, and Pietro Lio. DEFT: Efficient fine-tuning of diffusion models by learning the generalised $h$-transform. InThe Thirty-Eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  42. [43]

    Principled probabilistic imaging using diffusion models as plug-and-play priors

    Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Principled probabilistic imaging using diffusion models as plug-and-play priors. InThe Thirty-Eighth Annual Conference on Neural Information Processing Systems, 2024. 15

    2024

  43. [44]

    Boffi, and Eric Vanden-Eijnden

    Michael Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26(209):1–80, 2025

    2025

  44. [45]

    Albergo, Mark Goldstein, Nicholas M

    Michael S. Albergo, Mark Goldstein, Nicholas M. Boffi, Rajesh Ranganath, and Eric Vanden-Eijnden. Stochastic interpolants with data-dependent couplings, September 2024

    2024

  45. [46]

    Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. InThe Eleventh International Conference on Learning Representations, 2023

    2023

  46. [47]

    FlowDPS : Flow-driven posterior sampling for inverse problems

    Jeongsol Kim, Bryan Sangwoo Kim, and Jong Chul Ye. FlowDPS : Flow-driven posterior sampling for inverse problems. InProceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 12328–12337, October 2025

    2025

  47. [48]

    Exact Conditional Score-Guided Generative Modeling for Amortized Inference in Uncertainty Quantification, June 2025

    Zezhong Zhang, Caroline Tatsuoka, Dongbin Xiu, and Guannan Zhang. Exact Conditional Score-Guided Generative Modeling for Amortized Inference in Uncertainty Quantification, June 2025

    2025

  48. [49]

    Error estimates of a training-free diffusion model for high-dimensional sampling, January 2026

    Pengjun Wang, Zezhong Zhang, Minglei Yang, Feng Bao, Yanzhao Cao, and Guannan Zhang. Error estimates of a training-free diffusion model for high-dimensional sampling, January 2026. 16 Contents A Related work 17 A.1 Generative posterior samplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.2 Benchmarking vs. understanding . . . . . . . . . ...

    2026

  49. [50]

    the true posterior (where available), derived in Appendix D.1,

  50. [51]

    FSR: the finite-sample regime introduced in Section 4, 3.σ-DPS [16], defined in Equation (16) with unmodified prefactorσ −2 y , 4.ζ-DPS [16], defined in Equation (25) with modified prefactorζ/∥y− A(m 0|t(xt))∥2, 5.ΠGDM [17], defined in Equation (21), and

  51. [52]

    tail-event

    TMPD [18], defined in Equation (22). The FSR and DPS methods are present for all problems, whileΠGDM and TMPD are only applicable when the measurement model is affine. Heatmaps use power-law normalization with exponent γ= 0.55 . The color ceiling is set per row to the 99th percentile of that row’s analytic field with flipping at the99.9th percentile to pr...

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