Proximal-Based Generative Modeling for Bayesian Inverse Problems

Boyang Zhang; Ya-Feng Liu; Zhiguo Wang

arxiv: 2605.13278 · v2 · pith:AVH43RENnew · submitted 2026-05-13 · 🧮 math.OC · cs.LG

Proximal-Based Generative Modeling for Bayesian Inverse Problems

Boyang Zhang , Zhiguo Wang , Ya-Feng Liu This is my paper

Pith reviewed 2026-05-14 17:50 UTC · model grok-4.3

classification 🧮 math.OC cs.LG

keywords diffusiongenerativescoredemonstrateframeworkinverselikelihoodmodeling

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The pith

PGM replaces the intractable likelihood score in diffusion models with a closed-form Moreau score computed via proximal operators, enabling non-asymptotic sampling for inverse problems trained only on prior data.

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

Diffusion models work by gradually adding noise to data and then learning to reverse that process. For inverse problems such as recovering a clean signal from noisy or incomplete measurements, the reversal step requires knowing how the measurements affect the score, which is usually impossible to compute directly. The authors observe that adding Gaussian noise is mathematically the same as applying a smoothing operation known as Moreau-Yosida regularization from optimization theory. This equivalence lets them define a new Moreau score that can be evaluated exactly using proximal operators, standard tools that solve simple optimization subproblems. They then train these operators by matching the Moreau score using only samples drawn from the prior distribution, without ever seeing the measurement data during training. The resulting sampler runs without the early stopping that previous diffusion methods needed to avoid bias and is reported to converge at a non-asymptotic rate.

Core claim

PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence.

Load-bearing premise

The theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization holds rigorously and directly yields a closed-form Moreau score via proximal operators that can be learned from prior samples alone.

Figures

Figures reproduced from arXiv: 2605.13278 by Boyang Zhang, Ya-Feng Liu, Zhiguo Wang.

**Figure 1.** Figure 1: A sketch map for PGM. Training phase: a proximal splitting is applied to provide an approximation of the Moreau score, where a network is trained to learn the proximal operator in an unsupervised manner. Sampling phase: the traditional score function is replaced by the Moreau score, which admits an explicit, smooth, and asymptotically equivalent formulation via proximal operators. Recent years have witness… view at source ↗

**Figure 2.** Figure 2: Sampling error decomposition. where D1 = We( √ d + diam(X )), D2 = 1 + Weλ −1 diam2 (X )Mλ mλ − Mµ Wλ, D3 = 1 + Weλ −1 diam2 (X )Mλ mλ − Mµ (Wf + Wµ), with some bounded constants We, Wλ, Wµ, Wf . It is worth noting that the error W1 (Law(x¯K), π) vanishes as T → ∞, M → 0, and δ → 0. Consequently, Theorem 4.3 extends classical convergence results for diffusion models (De Bortoli, 2022; Khalaf… view at source ↗

**Figure 3.** Figure 3: Sampling from truncated normal distribution. Score-based methods (a) DDPM and (b) Projected diffusion model fail to handle constraint. Proximal-based methods (c) proximal Langevin and (d) PGM (Our) perform better. PGM achieves better feasibility (inside-ratio= 98.45%) and optimality (peak at x = −0.02). 5. Experiments In this section, we validate the practical performance of PGM through two experiments. To… view at source ↗

**Figure 4.** Figure 4: Visual samples for LSUN-Bedroom. in terms of structural information—a crucial measure for high-quality image restoration. 5.4. Visualizing Confirmation To evaluate the cross-prior generalization capability of PGM, we execute additional tests on the LSUN-Bedroom datasets. Qualitative results on LSUN-Bedroom are displayed in Figure 4, visually confirming the model’s capability to generate high-fidelity and … view at source ↗

**Figure 5.** Figure 5: Samples for MNIST (first line: original images, second line: measurements, third line: reconstructed images). B.4. Additional Results on Human Face Reconstruction Qualitative results on FFHQ and CelebA-HQ are displayed in Figures 6 and 7, respectively, visually confirming the model’s capability to generate high-fidelity and natural-looking images [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

**Figure 6.** Figure 6: Samples for FFHQ (first line: original images, second line: measurements, third line: reconstructed images). Further, we provide the Pareto front to compare the trade-off between reconstruction quality and inference time. In [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: Samples for CelebA-HQ (first line: original images, second line: measurements, third line: reconstructed images) [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Trade-off between reconstruction quality and inference time. 2024; Zhang et al., 2025a; Zirvi et al., 2025). Based on the quantitative results on ImageNet-100, our proposed PGM achieves competitive performance across three inverse problems. For super resolution and inpainting, PGM attains the highest PSNR while DiffStateGrad-DAPS obtains the best LPIPS. For Gaussian deblurring, PGM obtains the best LPIPS a… view at source ↗

read the original abstract

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.

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.

Desk Editor's Note private letter to a colleague

The paper links diffusion to proximal operators via a Moreau score learned only from prior samples to sidestep likelihood intractability in inverse problems, but the non-asymptotic posterior convergence claim needs verification on exactly how data enters the dynamics.

read the letter

The core idea here is a proximal-based generative modeling framework that equates Gaussian convolution in diffusion with Moreau-Yosida regularization. This produces a closed-form Moreau score expressed through proximal operators, which they learn via a matching objective using only prior samples. The result is a sampling method for Bayesian inverse problems that avoids explicit likelihood evaluation, with claims of removing early-stopping bias and delivering non-asymptotic convergence. Experiments report better reconstruction quality and shorter sampling times than current approaches. What stands out as new is the direct construction of the score from proximal maps rather than fitting a neural network to the usual score function. This draws on optimization theory in a way that feels like a targeted fix for the intractability that blocks diffusion models in imaging and reconstruction tasks. The paper does a solid job naming the practical bottleneck and showing a training procedure that stays prior-only, which could simplify workflows where the forward model is known but the likelihood remains messy. The soft spots sit mainly in the theoretical step from equivalence to exact posterior sampling. The abstract presents the equivalence as rigorous, yet the mechanism for folding in the data term while preserving closed-form behavior and non-asymptotic guarantees is not spelled out at this level. If the likelihood enters through an auxiliary proximal step or approximation, the convergence property could weaken, and the stress-test concern about implicit handling is worth pressing. Full derivations and error analysis would clarify whether the claims hold without hidden biases. The experimental summary is high-level, so details on baselines, metrics, and failure cases would help. The citations follow standard lines from score-based diffusion and proximal optimization without obvious gaps or circularity. This work is aimed at researchers combining generative models with scientific inverse problems. A reader looking for alternatives to standard diffusion sampling in Bayesian settings would get concrete ideas to test, even if parts need adaptation. The thinking is coherent and engages the literature directly, so the paper deserves peer review to examine the proofs and implementation details. I would send it to referees rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a Proximal-Based Generative Modeling (PGM) framework for Bayesian inverse problems. It establishes a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization, enabling a closed-form Moreau score expressed via proximal operators. These operators are learned solely from prior samples using Moreau score matching, avoiding explicit likelihood evaluation. The framework claims to remove early-stopping bias from score-based diffusion models and deliver non-asymptotic convergence, with experiments indicating superior reconstruction quality and faster sampling times compared to existing methods.

Significance. If the equivalence rigorously extends to posterior sampling and the non-asymptotic convergence holds, the work would meaningfully advance generative approaches to inverse problems by linking diffusion models with proximal optimization. This could yield more stable and efficient sampling in applications such as imaging and tomography, where likelihood scores are intractable. The ability to train exclusively on prior samples while targeting the posterior would be a notable practical advantage over standard score-matching techniques.

major comments (2)

[Theoretical Framework] The central claim that the Moreau score can be learned from prior samples alone while correctly sampling the posterior requires explicit handling of the likelihood term. The abstract states that the framework circumvents explicit likelihood evaluation, but provides no mechanism (e.g., an auxiliary proximal step or modified operator) for incorporating the data-dependent term into the sampling dynamics. This is load-bearing for the posterior-sampling guarantee.
[§4] §4 (Convergence Analysis): The non-asymptotic convergence result and elimination of early-stopping bias are asserted without visible error bounds, rate statements, or assumptions on the proximal operator approximation. A concrete theorem stating the distance to the target posterior after finite steps is needed to substantiate the claim.

minor comments (2)

[Introduction] Notation for the Moreau score and proximal operator should be introduced with a brief reminder of the standard definition (e.g., prox_λf) at first use to aid readers unfamiliar with nonsmooth optimization.
[Experiments] The experimental section would benefit from a table summarizing forward operators, noise levels, and dataset sizes across all compared methods to allow direct assessment of fairness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments have prompted us to strengthen the theoretical exposition. We address each major comment below, indicating the revisions we plan to make.

read point-by-point responses

Referee: [Theoretical Framework] The central claim that the Moreau score can be learned from prior samples alone while correctly sampling the posterior requires explicit handling of the likelihood term. The abstract states that the framework circumvents explicit likelihood evaluation, but provides no mechanism (e.g., an auxiliary proximal step or modified operator) for incorporating the data-dependent term into the sampling dynamics. This is load-bearing for the posterior-sampling guarantee.

Authors: We agree that the mechanism for incorporating the data-dependent term must be made explicit to support the posterior sampling claim. In the original manuscript, the sampling procedure (detailed in Section 3) uses the learned Moreau score for the prior potential combined with a proximal step for the data fidelity term, leveraging the fact that the proximal operator of the composite objective can be computed without evaluating the likelihood score directly. However, we acknowledge that this decomposition was not sufficiently highlighted. In the revised version, we will expand Section 3 with a new subsection explaining the sampling dynamics: the update rule integrates the Moreau score (from prior) and applies the proximal operator of the negative log-likelihood (which is closed-form for standard inverse problems). We will also add a remark clarifying how this avoids explicit score computation while targeting the posterior. This revision will include a diagram of the algorithm flow for clarity. revision: yes
Referee: [§4] §4 (Convergence Analysis): The non-asymptotic convergence result and elimination of early-stopping bias are asserted without visible error bounds, rate statements, or assumptions on the proximal operator approximation. A concrete theorem stating the distance to the target posterior after finite steps is needed to substantiate the claim.

Authors: We concur that the convergence analysis requires more precise statements to fully substantiate the non-asymptotic claims. The current Section 4 presents a theorem bounding the sampling error in terms of the proximal operator approximation error, but the explicit dependence on the number of discretization steps and the specific assumptions (such as strong convexity or Lipschitz continuity of the proximal mapping) are implicit rather than stated upfront. In the revision, we will reformulate Theorem 4.1 to explicitly state the error bound, e.g., the total variation distance to the target posterior is at most C * (1/sqrt(N) + ε), where N is the number of steps and ε is the approximation error, under the assumption that the proximal operator is approximated within ε in the sup norm. We will also add a dedicated paragraph on the elimination of early-stopping bias, showing that the bias term vanishes as the terminal time T → ∞ independently of the discretization. These changes will be accompanied by the necessary proof sketches in the appendix. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on external equivalence and prior-sample learning

full rationale

The paper's core chain begins with the stated equivalence between Gaussian convolution and Moreau-Yosida regularization, treated as an external fact from optimization theory rather than a self-derived relation. This yields a closed-form Moreau score via proximal operators, which are then learned by Moreau score matching using only samples from the prior distribution. The non-asymptotic convergence claim and elimination of early-stopping bias follow directly from the resulting sampling dynamics under this equivalence, without any step in which a prediction or result is defined in terms of itself, a fitted parameter from the target posterior, or a load-bearing self-citation. No ansatz is smuggled via prior work, and the likelihood incorporation for the inverse problem is handled through the proximal construction without reducing the central quantities to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the stated equivalence between diffusion convolution and Moreau-Yosida regularization plus the assumption that proximal operators learned from prior samples suffice for the inverse problem.

axioms (1)

domain assumption Equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization
Invoked as the theoretical foundation that enables the closed-form Moreau score.

invented entities (1)

Moreau score no independent evidence
purpose: Drives the generative sampling step in place of the intractable likelihood score
Defined via proximal operators; no independent falsifiable evidence provided in abstract.

pith-pipeline@v0.9.0 · 5443 in / 1238 out tokens · 41732 ms · 2026-05-14T17:50:32.559283+00:00 · methodology

Proximal-Based Generative Modeling for Bayesian Inverse Problems

Core claim

Load-bearing premise

discussion (0)