Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework

Rajesh Shrestha; Xiao Fu

arxiv: 2603.10281 · v3 · submitted 2026-03-10 · 💻 cs.LG · cs.AI· cs.CV

Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework

Rajesh Shrestha , Xiao Fu This is my paper

Pith reviewed 2026-05-15 12:44 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CV

keywords ADMMplug-and-playscore-based generative modelsinverse problemsconvergence analysisdenoiserfixed-point convergenceLangevin dynamics

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The pith

A three-stage AC-DC denoiser makes score-based models converge inside ADMM iterations for inverse problems.

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

The paper shows how to integrate score-based generative models as priors into the ADMM algorithm for solving inverse problems. It introduces an AC-DC denoiser that performs auto-correction by adding Gaussian noise, directional correction with conditional Langevin dynamics, and then score-based denoising. This corrects the mismatch between the manifolds used to train the score functions and the geometry of ADMM iterates that include dual variables. With proper parameter choices the full ADMM step becomes a weakly nonexpansive operator, which yields high-probability convergence to a small ball around a fixed point even with a constant step size. Under weaker conditions the denoiser is bounded and convergence holds with an adaptive step-size schedule, while experiments show improved solution quality over standard plug-and-play baselines.

Core claim

The central claim is that the proposed ADMM-PnP framework with the AC-DC denoiser ensures convergence: under proper denoiser parameters each ADMM iteration is a weakly nonexpansive operator, ensuring high-probability fixed-point ball convergence using a constant step size; under more relaxed conditions the AC-DC denoiser is bounded and yields convergence under an adaptive step size schedule.

What carries the argument

The AC-DC denoiser, a three-stage process of auto-correction via additive Gaussian noise, directional correction using conditional Langevin dynamics, and score-based denoising, which enforces the weakly nonexpansive or bounded property needed for ADMM convergence.

If this is right

Each full ADMM iteration remains weakly nonexpansive, so the sequence converges in high probability to a small ball around the fixed point with a fixed step size.
Convergence is guaranteed with an adaptive step-size schedule when the denoiser is only bounded.
Solution quality improves on a range of inverse problems compared with plug-and-play methods that lack the AC-DC stages.
Score-based generative priors can be used directly inside ADMM without causing divergence once the three-stage correction is applied.

Where Pith is reading between the lines

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

The AC-DC correction stages could be inserted into other proximal splitting methods that rely on denoisers or proximal maps.
The same three-stage taming approach may stabilize convergence when score models are used in non-convex optimization settings.
Practitioners can apply off-the-shelf score models to new inverse problems with reduced risk of divergence and less manual step-size tuning.

Load-bearing premise

The score-based denoiser after the AC and DC stages must satisfy the weakly nonexpansive or bounded property required by the convergence arguments.

What would settle it

Run the algorithm on a simple inverse problem with a score model whose output after AC-DC stages violates the weakly nonexpansive condition and check whether the iterates diverge or fail to enter a small ball around any fixed point.

read the original abstract

While score-based generative models have emerged as powerful priors for solving inverse problems, directly integrating them into optimization algorithms such as ADMM remains nontrivial. Two central challenges arise: i) the mismatch between the noisy data manifolds used to train the score functions and the geometry of ADMM iterates, especially due to the influence of dual variables, and ii) the lack of convergence understanding when ADMM is equipped with score-based denoisers. To address the manifold mismatch issue, we propose ADMM plug-and-play (ADMM-PnP) with the AC-DC denoiser, a new framework that embeds a three-stage denoiser into ADMM: (1) auto-correction (AC) via additive Gaussian noise, (2) directional correction (DC) using conditional Langevin dynamics, and (3) score-based denoising. In terms of convergence, we establish two results: first, under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator, ensuring high-probability fixed-point $\textit{ball convergence}$ using a constant step size; second, under more relaxed conditions, the AC-DC denoiser is a bounded denoiser, which leads to convergence under an adaptive step size schedule. Experiments on a range of inverse problems demonstrate that our method consistently improves solution quality over a variety of baselines.

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 wraps score denoisers in AC-DC stages to fit them into ADMM and claims two convergence results, but those results rest on operator properties that may not hold for general trained scores.

read the letter

The punchline is that this work gives a concrete way to make score-based generative models play nicely with ADMM by wrapping them in auto-correction and directional correction stages. That addresses the manifold mismatch from dual variables, which has been a sticking point. They do two things well. First, the AC-DC construction is new and seems tailored to the problem. Second, they prove two convergence statements: one for constant step size using weak nonexpansiveness, and one for adaptive steps using boundedness. That's more than most PnP papers offer. The experiments claim consistent gains over baselines on inverse problems. The soft spot is in the assumptions. The convergence relies on the AC-DC-score operator being weakly nonexpansive or bounded under proper parameters. There is no general argument that this holds for typical trained score networks across different data and geometries. If the paper only verifies it for specific cases or leaves it as a modeling choice, that limits how far the theory goes. The abstract does not detail the parameter conditions or proof sketches, so it is tough to judge if the results are tight or loose. This paper is for people working on inverse problems where they want to use diffusion priors inside classical optimizers. A reader interested in plug-and-play methods or score-based solvers would get value from the framework and the theory attempt. I would send it to peer review. The core idea is solid enough and the claims are worth checking against the full proofs and experiments.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes an ADMM plug-and-play (ADMM-PnP) framework that integrates score-based generative models as priors for inverse problems by introducing the AC-DC denoiser, consisting of auto-correction via additive Gaussian noise, directional correction using conditional Langevin dynamics, and score-based denoising. It establishes two convergence results: (1) under suitable denoiser parameters, ADMM iterations are weakly nonexpansive operators leading to high-probability fixed-point ball convergence with constant step size; (2) under relaxed conditions, the AC-DC denoiser is bounded, enabling convergence with an adaptive step size. Numerical experiments on various inverse problems show consistent improvements over baselines.

Significance. If the convergence guarantees hold for general score-based denoisers, this would provide a theoretically grounded integration of powerful generative priors into ADMM-style optimization, advancing plug-and-play methods for inverse problems in imaging and related domains.

major comments (1)

[Convergence analysis (as stated in the abstract and theoretical sections)] The two convergence results (weak nonexpansiveness for constant-step high-probability ball convergence, and boundedness for adaptive-step convergence) require that the composite AC-DC-score operator satisfies the stated contraction or boundedness properties. No general argument is supplied showing that a trained score network, after the specific AC additive-noise and DC conditional-Langevin stages, inherits these properties for arbitrary data manifolds and dual-variable geometries; the claims therefore rest on an unverified model- and problem-dependent assumption.

minor comments (1)

[Experiments] The experimental section would benefit from explicit reporting of the exact inverse problems, datasets, and quantitative metrics used to demonstrate gains over baselines.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment on the convergence analysis below.

read point-by-point responses

Referee: [Convergence analysis (as stated in the abstract and theoretical sections)] The two convergence results (weak nonexpansiveness for constant-step high-probability ball convergence, and boundedness for adaptive-step convergence) require that the composite AC-DC-score operator satisfies the stated contraction or boundedness properties. No general argument is supplied showing that a trained score network, after the specific AC additive-noise and DC conditional-Langevin stages, inherits these properties for arbitrary data manifolds and dual-variable geometries; the claims therefore rest on an unverified model- and problem-dependent assumption.

Authors: We agree that the two convergence theorems are conditional on the composite AC-DC-score operator satisfying weak nonexpansiveness (under suitable parameters for constant-step high-probability ball convergence) or boundedness (for adaptive-step convergence). The AC and DC stages are explicitly designed to address manifold mismatch and dual-variable geometry by injecting controlled noise and performing conditional Langevin correction, thereby enabling the composite operator to meet these properties for appropriately chosen parameters. We do not claim a universal, parameter-free guarantee that holds for every trained score network on arbitrary manifolds; such a result would be unrealistic. Our contribution lies in showing how the AC-DC construction makes the required properties achievable in practice, with the analysis deriving explicit conditions on the parameters. We will revise the abstract, Section 3, and Section 4 to more explicitly state these assumptions and the role of parameter selection, and we will add a brief remark clarifying the model-dependent nature of the guarantees. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives two conditional convergence results for ADMM-PnP by showing that the composite AC-DC-score operator is weakly nonexpansive (constant-step case) or bounded (adaptive-step case) when denoiser parameters are chosen appropriately. These operator properties are established from the explicit three-stage construction rather than by redefining the target result in terms of itself or by renaming a fitted quantity as a prediction. No load-bearing self-citation, uniqueness theorem, or ansatz is invoked to close the argument; the claims remain conditional on verifiable properties of the proposed denoiser and are therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard assumptions from optimization theory (nonexpansiveness of certain operators) and generative modeling (score functions approximating gradients of log-density); the AC-DC construction itself is an invented entity whose properties are asserted rather than derived from first principles.

axioms (1)

domain assumption Score-based denoisers can be rendered weakly nonexpansive or bounded after AC and DC corrections under suitable parameter choices
Invoked to obtain the two convergence guarantees stated in the abstract

invented entities (1)

AC-DC denoiser no independent evidence
purpose: Three-stage module that corrects manifold mismatch before applying a score-based denoiser inside ADMM
Newly proposed construction consisting of auto-correction, directional correction, and score-based denoising stages

pith-pipeline@v0.9.0 · 5547 in / 1328 out tokens · 41071 ms · 2026-05-15T12:44:46.149841+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator... AC-DC denoiser is a bounded denoiser
Foundation/AbsoluteFloorClosure absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Assumption 2 (Smoothness of log p_data)... M-smooth... Assumption 3 (Coercivity for -log p_data)

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