Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach

arXiv:2506.03979 · cs.LG, cs.CV, cs.NA, eess.IV, math.NA, stat.ML · Submitted 2025-06-04 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Solving Inverse Problems via Diffusion-Based Priors".

Jane: Here's the gist of the paper. The first text is an abstract/summary of a novel method called Approximation-Free Diffusion Posterior Sampler (AFDPS), while the second text appears to be a list of references and titles related to ensemble Kalman inversion, sequential Monte Carlo (SMC) methods, and convergence analysis in high dimensions.

Tom: My task is to synthesize these into one long, detailed summary of the paper described by the first text, leveraging the context provided by the second.

Jane: Here is my detailed synthesis: Detailed Research Summary: Approximation-Free Diffusion Posterior Sampling (AFDPS) for Bayesian Inverse Problems via Diffusion Models This research introduces a novel framework, termed Approximation-Free Diffusion Posterior Sampler (AFDPS), designed to solve complex Bayesian Inverse Problems (BIPs) by leveraging the generative power of Diffusion Models (DMs) as the prior distribution.

Tom: The core innovation lies in achieving exact posterior sampling *without* relying on heuristic approximations typically employed in existing DM-based methods.

Title and authors: Tom: So we’re looking at this new paper, "Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach." It sounds like they’re tackling a big problem in figuring out what the world looks like when you only have incomplete data.

Jane: Exactly. The title tells us the core idea is using diffusion models as priors to solve inverse problems, but with a specific focus on doing it exactly, without those usual shortcuts people take.

Lu: Right, so what they’re proposing here is an ensemble-based algorithm that samples the posterior distribution without relying on those common heuristic approximations that are often used in existing diffusion model based methods.

Meng: If I’m honest, when I look at this, my first thought is always about how much computation this will actually take to run on real traffic. It sounds theoretically clean, but does it translate to something we can deploy right now?

Tom: Well, the paper explains that they've derived a novel partial differential equation that precisely governs how the posterior distribution evolves as it’s shaped by this evolving diffusion prior. This is a big step because it moves away from just using whatever approximation works best.

Jane: That PDE has two main parts, actually: one handles the dynamics coming from the pre-trained score function in the diffusion process, and another term that manages the reweighting needed for sampling accuracy.

Lu: And they’ve built a stochastic weighted particle method on top of this PDE to simulate it. They show they can get asymptotic convergence to these derived PDE dynamics in terms of mean-field consistency in the two-Wasserstein sense, which is a strong theoretical guarantee for how the particle system behaves over time.

Meng: Convergence to a PDE sounds nice on paper, but what about the practical error? How much error are we actually looking at when we run this thing on something like an image?

Tom: They did derive some pretty tight bounds on that. The total variation distance between the sampled distribution and the true posterior is bounded by something like C times y times square root of m two epsilon s, where m is the number of particles and epsilon s is related to the training error of that pre-trained score function.

Jane: So, if we have a good prior from a diffusion model, we can control the error just by adjusting how many particles we use or how long we run the process over time. That’s a concrete way to manage uncertainty.

Title and authors: Lu: They also found an optimal trade-off controlled by a time horizon T between the mismatch of the prior and the approximation error of the score function, optimizing this relationship via m two two T squared equals T squared epsilon s, leading to an upper bound on total variation distance that looks like TV(qby,T, qy,zero) <= C(two) y r m2 squared T squared + T squared epsilon s.

Meng: That formula with the time horizon and the particle count tells me we have to balance two things: how much we trust our prior model versus how many particles we need to track the true distribution. If I have a massive image, that number of particles could get really big, making it slow.

Tom: The authors show that in the many-particle limit, as the number of particles goes to infinity, the expected Wasserstein distance between the particle distribution and the true posterior vanishes for any fixed time horizon T, confirming asymptotic convergence.

Jane: That sounds reassuring for us because it means we don't have to worry about whether a finite run is "good enough," provided we let it run long enough. But what about how we actually use this in practice?

Lu: They present two algorithmic variants for implementation: one using a stochastic differential equation, the AFDPS-SDE variant, and another using a probability flow ordinary differential equation augmented with a stochastic corrector step.

Tom: The SDE variant is noted for producing visually compelling and intricate reconstructions because it excels at generating sharp details and fine textures, with superior LPIPS scores on datasets like ImageNet-two hundred fifty-six where they hit zero point three nine two five.

Jane: And the ODE variant seems to focus more on smoothness. They say it achieves competitive performance in terms of PSNR on datasets like FFHQ, getting a score of twenty-four point nine eight PSNR for that task.

Meng: So, if I need a really sharp image, I’d probably lean toward the SDE approach based on those LPIPS numbers. But then we have to consider the trade-off with the ODE variant when smoothness is more important for our application.

Lu: They also analyzed the relationship between these two approaches and pointed out that a specific drift term involving V(t) squared gradient of x mu y is critical for guiding the sampler toward regions of high likelihood, which they showed tuning a parameter eta lets you move between FK Corrector dynamics when eta is zero and AFDPS dynamics when it’s one.

Title and authors: Tom: That specific term seems to be the key mechanism that actually steers the sampler correctly, not just some mathematical artifact of using diffusion models as priors. It shows the PDE structure is practically tunable for better sampling behavior.

Jane: So, we've seen them tackle a major hurdle: getting exact posterior sampling without relying on those common heuristic approximations in diffusion model based inverse problems. But what does this mean for the real world application of reconstructing things from limited data?

Lu: It means that if you use a diffusion model as your prior, you can get a mathematically rigorous framework to sample from the resulting complex distribution, which is much better than just throwing a heuristic method at it.

Meng: For us in production, this means we need to be very careful about how we set up our initial diffusion prior and how long we let the sampling process run to keep those error bounds tight on real data.

Tom: Ultimately, this paper, "Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach," provides a mathematically rigorous pathway to posterior sampling in complex inverse problems using diffusion models as priors.

Jane: It bridges the gap between the generative power of diffusion models and the exact dynamics needed for accurate Bayesian inference by deriving that novel PDE that governs evolution.

Lu: The combination of rigorous theoretical guarantees, like those convergence proofs and error bounds, alongside superior empirical performance across FFHQ-two hundred fifty-six and ImageNet-two hundred fifty-six positions this as a powerful tool for high-fidelity reconstruction in complex inverse problems <ref:2506.03979#pg3>.

Meng: It’s a solid methodology to have on the table when we are trying to move from simulation to actual deployment where we need reliable results.

Lalam: I can confirm that the paper successfully moves beyond relying on heuristic approximations by deriving an exact partial differential equation that precisely characterizes this posterior evolution, avoiding those common approximations.

Tom: So, if you’re driving or cooking and you need a quick summary of what this means for us right now, remember it’s about getting exact posterior sampling without those tricky shortcuts.

Jane: Right, and as we wrap up today's discussion on this paper, we've seen how the AFDPS framework offers flexibility between SDE and ODE implementations for different needs.

The paper's summary: Tom: So we’re looking at this paper "Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach" and what they’re doing with diffusion models as priors to solve these inverse problems, but focusing on getting an exact answer without those usual shortcuts.

Jane: Right, so the big idea is they’ve figured out a way to use diffusion models as the starting point for sampling the posterior distribution in complex inverse problems, but they are deriving a partial differential equation that precisely governs how that distribution evolves.

Lu: Exactly. They aren't just using some existing approximation; they are deriving this new PDE itself, which has a modified diffusion term and a reweighting term to keep the sampling accurate.

Meng: So it’s moving away from just picking the best heuristic, which is exactly what we need when we’re trying to ship something reliable at scale. What does that novel PDE actually look like in practice?

Tom: It's characterized by two main things: one part handles the dynamics coming from the pre-trained score function encoded in the diffusion process, and another term manages the reweighting needed during sampling to stay true to the actual distribution.

Jane: And they simulate this PDE using a stochastic weighted particle method, which is their backbone for both their SDE and ODE versions of the algorithm.

Lu: The theory here is pretty heavy lifting; they provide convergence guarantees showing that as you run the ensemble algorithm long enough, it actually converges to the dynamics defined by that derived PDE in terms of mean-field consistency in the two-Wasserstein sense.

Meng: That sounds mathematically sound, but what about error? If I’m running this on a real medical scan or an industrial sensor reading, how much error am I going to see?

Tom: They did derive some pretty tight bounds on that total variation distance between the sampled distribution and the true posterior, which is bounded by something like C times y times square root of m two epsilon s.

The paper's summary: Jane: So it means the accuracy depends directly on how many particles you use—that’s m—and how well the original diffusion model was trained—that’s epsilon s.

Lu: They also found an optimal trade-off controlled by a time horizon T between how much the prior mismatches and how much error is in approximating that score function, optimizing it through m two two T squared equals T squared epsilon s.

Meng: That formula with the time horizon and particle count tells me we have to balance trusting our prior model against how many particles we need to track the true distribution. If I have a massive image, that number of particles could get really big, making it slow.

Tom: They also showed that in the many-particle limit—when n goes to infinity—the expected Wasserstein distance between the particle distribution and the true posterior goes to zero for any fixed time horizon T, confirming asymptotic convergence.

Jane: That sounds reassuring because it means we don't have to worry about whether a finite run is "good enough," provided we let it run long enough. But they also showed two ways to do this: an SDE variant and an ODE variant with a corrector step.

Lu: The SDE variant is noted for producing visually compelling reconstructions because it excels at generating sharp details and fine textures, with superior LPIPS scores on datasets like ImageNet-two hundred fifty-six where they hit zero point three nine two five.

Tom: And the ODE variant seems to focus more on smoothness; they say it achieves competitive performance in terms of PSNR on tasks like FFHQ, getting a score of twenty-four point nine eight PSNR for that task.

Jane: So if I need a really sharp image, I’d probably lean toward the SDE approach based on those LPIPS numbers. But then we have to consider the trade-off with the ODE variant when smoothness is more important for our application.

Lu: They also analyzed how these two approaches relate and pointed out that a specific drift term involving V(t) squared gradient of x mu y is critical for guiding the sampler toward regions of high likelihood, which they showed tuning a parameter eta lets you move between FK Corrector dynamics when eta is zero and AFDPS dynamics when it’s one.

The paper's summary: Meng: That specific term seems to be the key mechanism that actually steers the sampler correctly, not just some mathematical artifact of using diffusion models as priors. It shows the PDE structure is practically tunable for better sampling behavior.

Tom: So we've seen them tackle a major hurdle: getting exact posterior sampling without relying on those common heuristic approximations in diffusion model based inverse problems.

Jane: But what does this mean for us when we actually use this to reconstruct things from limited data?

Lu: It means if you use a diffusion model as your prior, you can get a mathematically rigorous framework to sample from the resulting complex distribution, which is much better than just throwing a heuristic method at it.

Tom: Ultimately, this paper provides a mathematically rigorous pathway to posterior sampling in complex inverse problems using diffusion models as priors by deriving that novel PDE that governs evolution.

Jane: It really bridges the gap between the generative power of diffusion models and the exact dynamics needed for accurate Bayesian inference.

Meng: It’s a solid methodology to have on the table when we are trying to move from simulation to actual deployment where we need reliable results.

The paper's improvements: Tom: So we’re looking at how they suggest making this AFDPS method even better and what that means for its practical use in inverse problems.

Jane: The paper suggests two main ways to improve things: first, by using the SDE version or the ODE version depending on what you actually need for your problem.

Lu: That makes sense because the SDE variant is good for generating sharp details and fine textures, which is great for imaging tasks like super-resolution.

Meng: And the ODE variant seems to focus more on smoothness, which they say gives competitive performance in PSNR on things like FFHQ. So it’s a trade-off based on what you need the output to look like.

Tom: They also highlighted that tuning that parameter eta lets you switch between dynamics, where eta equals zero is FK Corrector and when it's one, you get the AFDPS dynamics.

Jane: And they point out that this specific drift term involving V(t) squared gradient of x mu y is critical because it’s what guides the sampler toward regions of high likelihood in a way that makes sense for sampling.

Lu: Exactly, it’s not just some mathematical artifact; it shows that the derived PDE structure is practically tunable for better sampling behavior, which would make the algorithm much more flexible.

Meng: So if we can tune it to steer the sampler correctly, that reduces our reliance on just running a single fixed version of the algorithm and makes deployment more robust across different types of data.

Tom: They also showed how this structure helps control error bounds tightly, using that time horizon T and particle count M in their formula.

Jane: So, if you know how to tune it to steer the sampler right while keeping those error bounds tight with M and T, you get a very predictable level of accuracy for your reconstruction.

Lu: The core improvement here is moving from a fixed approach to a tunable one that respects the underlying physics of the posterior evolution better.

Meng: For me, that tunability is what matters at scale; we don't want to run massive ensembles if we can find a parameter setting that gives us enough accuracy with fewer particles.

Tom: So they’re essentially showing us how to take this theoretical framework and make it practically flexible for real-world engineering constraints.

Jane: It really bridges the gap between the theory and the practice of using these complex diffusion models to get reliable results from noisy data.

Conclusion: Tom: So we’re wrapping up on "Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach," summarizing what this paper actually achieves and why it matters for us as engineers.

Jane: The core message is that they’ve taken diffusion models, which are great at generating data, and used them to create a method that samples the true underlying distribution without any of those messy approximations we usually have to use.

Lu: They achieved this by deriving a specific partial differential equation that precisely describes how the posterior distribution evolves as it's shaped by that diffusion prior.

Meng: So, what does that mean for us in production? Does this mean we can finally trust the results from these diffusion-based inverse problems when we deploy them?

Tom: It means they have rigorous theoretical guarantees. They show convergence to the derived PDE dynamics in terms of mean-field consistency in the two-Wasserstein sense, which is a strong mathematical statement about how accurate it gets over time.

Jane: And they gave us concrete error bounds, tying the accuracy directly to how many particles you use and how well you trained your initial diffusion model score function.

Lu: They also showed that as you run the particle system for an infinite number of steps, the empirical distribution converges to the solution of that posterior PDE in a two-Wasserstein sense.

Meng: That convergence proof is solid, but I still have to worry about the cost. If we need a lot of particles M to keep that error low on a big dataset, it could be too slow for real-time applications.

Tom: That’s where the flexibility comes in; they offer two algorithmic variants: an SDE formulation and an ODE with a corrector step.

Jane: So you can pick the SDE if you want those sharp, intricate details, or the ODE if smoothness is more important for your specific use case.

Lu: And they’ve shown that tuning a parameter eta lets you transition between these dynamics, where eta equals zero is FK Corrector and one gives AFDPS dynamics.

Meng: That tuning capability is what I’m interested in; it means we can optimize the sampling behavior for our specific data without having to re-implement the whole framework every time.

Tom: In short, this paper provides a mathematically rigorous pathway to posterior sampling in complex inverse problems using diffusion models as priors.

Jane: It successfully bridges the gap between those powerful generative diffusion models and the exact dynamics required for accurate Bayesian inference.

Lu: The combination of theoretical convergence proofs and superior empirical performance across datasets like FFHQ-two hundred fifty-six positions this as a powerful tool for high-fidelity reconstruction in complex inverse problems.

Meng: It’s a solid methodology to have on the table when we are trying to move from simulation to actual deployment where we need reliable results.

Lalam: I am confident that the AFDPS framework provides the necessary mathematical foundation to sample these distributions accurately, and my analysis confirms its robustness under the conditions they set out.

Tom: We’ve seen how they tackle this major hurdle: getting exact posterior sampling without relying on those common heuristic approximations in diffusion model based inverse problems.

Jane: This work opens up clear avenues for extending this framework to other types of inverse problems and different diffusion architectures.

Lu: They laid a very strong foundation for how we can use generative models to drive the inference process, which is really ambitious research.

Meng: I’m interested to see if they can handle data that isn't clean or perfectly structured like these benchmarks.

Tom: Next week, we’ll look at a different paper on how vision-language models actually understand movies and what those benchmarks are showing us there.

Haoxuan Chen, Yinuo Ren, Martin Renqiang Min, Lexing Ying, Zachary Izzo

Stanford University · NEC Labs America

cs.LG, cs.CV, cs.NA, eess.IV, math.NA, stat.ML

Submitted: 2025-06-04

Updated: 2026-10-17

Comments: Accepted by TMLR 2026 (https://openreview.net/forum?id=qN8ASsfjKs)

Code: https://github.com/HaoxuanSteveC00/AFDPS-TMLR

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 86/100

The gist: The first text is an abstract/summary of a novel method called Approximation-Free Diffusion Posterior Sampler (AFDPS), while the second text appears to be a list of references and titles related to

Key concepts

Diffusion Models (DMs): DMs are generative models trained to gradually add noise to data until it becomes pure noise. In this context, they serve as a powerful prior distribution for the inverse problem, helping to guide the sampling process toward plausible solutions in high-dimensional spaces.
Partial Differential Equation (PDE) Derivation: The core innovation is deriving a precise PDE that models how the posterior distribution changes as it is shaped by the diffusion prior. This replaces heuristic approximations with a mathematically rigorous framework for tracking the true posterior dynamics.
2-Wasserstein Sense Convergence: This is a mathematical measure used to prove that an ensemble of samples converges to the true posterior distribution. Establishing convergence in this sense guarantees that, in the limit of many particles, the sampled distribution accurately reflects the true underlying probability density.
AFDPS Variants (SDE vs. ODE): The framework offers two algorithmic choices: an SDE variant for visually sharp reconstructions and an ODE+Corrector variant for smooth results. Tuning parameters allows practitioners to select the dynamic that best suits their specific inverse problem needs.

Terminology

Inverse Problem — 3× in this paper · explained in 2 episodes
Score Function — 2× in this paper · explained in 4 episodes
Diffusion Model — 1× in this paper · explained in 10 episodes
Diffusion Models — 1× in this paper · explained in 9 episodes
Diffusion Process — 1× in this paper · explained in 2 episodes
Sequential Monte Carlo (SMC) — 1× in this paper · explained in 2 episodes

Summary

The first text is an abstract/summary of a novel method called Approximation-Free Diffusion Posterior Sampler (AFDPS), while the second text appears to be a list of references and titles related to ensemble Kalman inversion, sequential Monte Carlo (SMC) methods, and convergence analysis in high dimensions.

My task is to synthesize these into one long, detailed summary of the paper described by the first text, leveraging the context provided by the second.

Here is my detailed synthesis:

Detailed Research Summary: Approximation-Free Diffusion Posterior Sampling (AFDPS) for Bayesian Inverse Problems via Diffusion Models

This research introduces a novel framework, termed Approximation-Free Diffusion Posterior Sampler (AFDPS), designed to solve complex Bayesian Inverse Problems (BIPs) by leveraging the generative power of Diffusion Models (DMs) as the prior distribution. The core innovation lies in achieving exact posterior sampling without relying on heuristic approximations typically employed in existing DM-based methods.

Core Methodology and Theoretical Foundation

The fundamental motivation stems from combining DM-based methods with Sequential Monte Carlo (SMC) techniques, drawing inspiration from established convergence analyses of ensemble Kalman inversion and sequential MCMC methods (as suggested by the cited literature [260]–[267]). The authors move beyond heuristic approximations by deriving a novel Partial Differential Equation (PDE) that precisely governs the evolution of the posterior distribution as it is shaped by an evolving diffusion prior.

This derived PDE is characterized by two key components:

A modified diffusion term: Reflecting the dynamics imposed by the pre-trained score function encoded within the DM process.
A reweighting term: Accounting for the necessary adjustments during posterior sampling to maintain accuracy relative to the true distribution.

This PDE is simulated using a stochastic weighted particle method, which forms the backbone of Algorithm 2 (for AFDPS-SDE) and Algorithm 4 (for AFDPS-ODE). The theoretical rigor is exceptionally high:

Convergence Guarantee: The authors provide comprehensive theoretical guarantees demonstrating that the ensemble-based algorithm converges asymptotically to the derived PDE dynamics. This convergence is established in terms of mean-field consistency in the 2-Wasserstein sense, a measure relevant to the evolution of probability measures, aligning with concepts explored in related literature [265].
Error Bounds: Crucially, they derive precise error bounds relating the posterior sampling accuracy to two key factors: the training error of the pre-trained score function and the number of particles in the ensemble. They also establish a trade-off controlled by a time horizon T between prior mismatch and score approximation error, optimizing this relationship via m 22T squared = T 2 epsilon s, leading to an optimal upper bound on the total variation distance: TV(q by, T, q y, 0) C(2)y sqrt m 2 epsilon s.
Limit Analysis: The proof concludes by showing that in the many-particle limit (N to infinity), the expected Wasserstein distance between the particle distribution and the true posterior vanishes for any fixed time horizon T, confirming asymptotic convergence.

Algorithmic Variants: SDE vs. ODE+Corrector

The AFDPS framework offers two distinct algorithmic implementations, allowing flexibility based on computational preference:

AFDPS-SDE Variant: This variant implements the diffusion model dynamics via a Stochastic Differential Equation (SDE) formulation. It is noted for producing visually compelling and intricate reconstructions, excelling in generating sharp details and fine textures, as evidenced by its superior LPIPS scores on datasets like ImageNet-256 (0.3925).
AFDPS-ODE Variant: This variant utilizes a Probability Flow Ordinary Differential Equation (ODE) implementation, augmented with a stochastic corrector step inspired by predictor-corrector schemes. This approach tends to yield results characterized by notable smoothness, achieving competitive performance in terms of PSNR on datasets like FFHQ (24.98 PSNR).

The authors also analyze the relationship between these two approaches, noting that the specific drift term involving V(t) 2 grad x mu y is critical. Tuning a parameter eta allows practitioners to transition between dynamics: eta=0 corresponds to FK Corrector dynamics, while eta=1 yields the AFDPS dynamics. The empirical results confirm that this specific term plays a vital role in guiding the sampler toward regions of high likelihood, thus enhancing overall performance.

Empirical Validation and Comparative Analysis

The efficacy of AFDPS is rigorously validated through extensive empirical testing across multiple imaging inverse problems utilizing large-scale datasets, including FFHQ-256 [60] and ImageNet-256 [61]. The results consistently demonstrate that both AFDPS variants outperform existing state-of-the-art DM-based posterior sampling methods, such as DPS, DCDP, SGS-EDM, FK-Corrector, and PF-SMC-DM.

Specifically:

Superior Metrics: AFDPS methods frequently surpass baselines in both PSNR and LPIPS metrics. For instance, AFDPS-SDE achieved the best LPIPS score on ImageNet-256 (0.3925) and a competitive LPIPS on FFHQ-256 (0.2580).
Qualitative Results: The authors emphasize that the specific dynamics allow for tuning parameters (eta) to control the influence of certain terms, proving that the derived PDE structure is not just theoretically sound but also practically tunable for optimal sampling behavior.

Conclusion and Significance

In summary, AFDPS represents a significant methodological advancement by providing a mathematically rigorous, approximation-free pathway to posterior sampling in DIP problems using DMs as priors. The paper successfully bridges the gap between the generative capabilities of diffusion models and the exact dynamics required for accurate Bayesian inference by deriving a novel PDE that precisely characterizes this evolution. The combination of rigorous theoretical guarantees (convergence proofs, error bounds) and superior empirical performance across challenging imaging tasks positions AFDPS as a powerful new tool for high-fidelity reconstruction in complex inverse problems. The work opens clear avenues for extending this framework to other inverse problems and various DM architectures.

Improvements for AI systems

Diffusion-based posterior sampling will eliminate heuristic approximations by deriving an exact partial differential equation governing posterior evolution, avoiding the usage of such approximations. This allows for more principled exploration of high-dimensional, multi-modal distributions compared to existing methods.
The AFDPS algorithm can be implemented using either an SDE approach or an ODE+Corrector approach, offering flexibility based on the specific inverse problem dynamics. For instance, the SDE implementation (Algorithm 2) is suited for generating visually compelling outputs with sharp details in imaging tasks like Gaussian Deblurring and Super-Resolution.
The AFDPS-ODE variant can be utilized for producing results characterized by notable smoothness, as demonstrated on tasks like Box Inpainting where it achieves best LPIPS on both datasets (FFHQ: 0.1969, ImageNet: 0.2716).
The theoretical framework provides a rigorous error bound for posterior estimation in terms of the training error of the pre-trained score function, quantified by TV(qby,T, qy,0) ≤ C(2) y r m2 squared T squared + T squared epsilon squared s.
The ensemble-based method's convergence in the many-particle limit is theoretically justified via propagation of chaos arguments, establishing that the empirical distribution of the particle system converges to the solution of the posterior PDE in a 2-Wasserstein sense.

Abstract

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing work that combines DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior and the empirical distribution of the generated samples can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods. Our code is available at the following Github repository.

Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach

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