arxiv: 2605.05387 · v1 · submitted 2026-05-06 · 💻 cs.LG · cs.IT· math.IT

Recognition: 3 theorem links

· Lean Theorem

Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees

Ahmad Aghapour , Erhan Bayraktar , Asaf Cohen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:41 UTC · model grok-4.3

classification 💻 cs.LG cs.ITmath.IT

keywords conditional diffusionlinear constraintsLangevin dynamicsinformation-theoretic boundszero-shot samplinginpaintingsuper-resolutionscore decomposition

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The pith

Replacing the unknown tangent score with its unconditional counterpart in diffusion sampling adds an error bounded only by the conditional mutual information between observed and unobserved signal components.

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

The paper shows that, under Gaussian noising, the score function splits into an observed-direction part that is fully determined by the linear measurement and a tangent part that remains unknown. Approximating the tangent part by the unconditional score produces a total error that decomposes into an initialization term and a pathwise mismatch term, each controlled by a dimension-free conditional mutual information. This decomposition explains why simple projection methods can bias samples and motivates a projected-Langevin start followed by guided denoising. The result matters for zero-shot conditional generation on tasks such as inpainting and super-resolution, where retraining a model for every new constraint is impractical.

Core claim

For Gaussian noising, the observed-direction score is exactly determined by the measurement; only the tangent conditional score is unknown. The error incurred by replacing this score by the unconditional tangent score is upper bounded by a dimension-free conditional mutual information between observed and unobserved components. This supplies an information-theoretic decomposition of the total sampling error into initialization error and pathwise score-mismatch error.

What carries the argument

Normal-tangent decomposition of the score function, which isolates the exactly known measurement-determined normal component from the unknown tangent conditional score under the Gaussian forward process.

If this is right

Projected-Langevin initialization followed by guided reverse denoising produces conditional samples with lower bias than pure projection-based baselines on inpainting and super-resolution tasks.
The information-theoretic bound separates initialization error from pathwise mismatch, allowing targeted improvements to each component.
The dimension-free character of the mutual-information bound implies that the guarantee does not degrade with ambient dimension.
Any sampler that reduces the tangent-score mismatch or improves mixing from the projected initialization will inherit the same decomposition.

Where Pith is reading between the lines

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

The same normal-tangent split may be useful for analyzing score-based samplers in other partially observed linear settings beyond diffusion.
Numerical verification of the bound on synthetic data with known mutual information would provide a direct test of the theory.
The framework suggests combining the initialization with other guidance techniques to further tighten the pathwise error term.
If similar decompositions exist for non-Gaussian forward processes, the approach could extend beyond the current Gaussian assumption.

Load-bearing premise

The forward noising process must be exactly Gaussian so that the observed-direction score is fully determined by the linear measurement.

What would settle it

Compute the exact conditional mutual information on a low-dimensional linear-Gaussian example, run the proposed sampler, and check whether the observed total variation or KL error to the true conditional distribution stays within the predicted bound.

Figures

Figures reproduced from arXiv: 2605.05387 by Ahmad Aghapour, Asaf Cohen, Erhan Bayraktar.

**Figure 1.** Figure 1: Three-point mixture in R 2 . (a) Unconstrained PF-ODE reproduces the prior mixture. (b) Naive projection-based guidance accumulates tangent drift dominated by the high-mass (0, 5) mode, biasing the conditional outcome. (c) Our projected Langevin initialization at t ∗ followed by the surrogate reverse dynamics recovers constraint-consistent samples with the correct mode structure. Step 1: Initialization for… view at source ↗

**Figure 2.** Figure 2: Visual overview of the proposed sampling process. Step 1: diffuse the constrained view at source ↗

**Figure 3.** Figure 3: Visual comparison for inpainting. DDNM can produce texture artifacts or seman view at source ↗

**Figure 4.** Figure 4: 8× super-resolution on ImageNet. DDNM tends to produce blurred or structurally inconsistent outputs, while LCDM-BAOAB recovers sharper edges and more realistic highfrequency detail. 18 view at source ↗

read the original abstract

We study zero-shot conditional sampling with pretrained diffusion models for linear inverse problems, including inpainting and super-resolution. In these problems, the observation determines only part of the unknown signal. The remaining degrees of freedom must be sampled according to the correct conditional data distribution. Existing projection-based samplers enforce measurement consistency by correcting the observed component during reverse diffusion. However, measurement consistency alone does not determine how probability mass should be distributed along the feasible set, and this can lead to biased conditional samples. We analyze this issue through a normal--tangent decomposition of the score function. For Gaussian noising, the observed-direction score is exactly determined by the measurement; only the tangent conditional score is unknown. We prove that the error from replacing this score by the unconditional tangent score is upper bounded by a dimension-free conditional mutual information between observed and unobserved components. This gives an information-theoretic decomposition into initialization and pathwise score-mismatch errors. Motivated by the theory, we propose a projected-Langevin initialization followed by guided reverse denoising, which outperforms a strong projection-based baseline in inpainting and super-resolution experiments.

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 derives a dimension-free conditional-MI bound on tangent-score mismatch for Gaussian diffusion under linear constraints and uses it to motivate a projected-Langevin initialization that beats pure projection baselines.

read the letter

The main new piece is the normal-tangent decomposition of the score under Gaussian noising. Only the tangent component is unknown once the linear measurement is given, and the error incurred by replacing the conditional tangent score with the unconditional one is bounded by the conditional mutual information between observed and unobserved parts. The bound is dimension-free and immediately yields the split into initialization error plus accumulated pathwise mismatch that the authors use to justify their sampler.

Referee Report

2 major / 2 minor

Summary. The paper analyzes zero-shot conditional sampling from pretrained diffusion models for linear inverse problems such as inpainting and super-resolution. It introduces a normal-tangent decomposition of the score under Gaussian noising, proves that the error incurred by replacing the conditional tangent score with the unconditional one is upper-bounded by a dimension-free conditional mutual information between observed and unobserved components, and obtains an information-theoretic decomposition of total error into initialization and pathwise mismatch terms. Motivated by the analysis, the authors propose a projected-Langevin initialization followed by guided reverse denoising and report improved performance over a strong projection-based baseline on inpainting and super-resolution tasks.

Significance. If the central bound holds, the work supplies a clean information-theoretic explanation for the bias of pure projection methods and a principled route to reduce pathwise error. The dimension-free character of the mutual-information bound and the explicit separation of initialization versus pathwise contributions are notable strengths that could inform both theory and algorithm design for constrained diffusion sampling.

major comments (2)

[§3] §3 (normal-tangent decomposition and Theorem 1): the proof that the observed-direction score is exactly recovered from the linear measurement appears to rely on the forward process being exactly Gaussian; the manuscript should state explicitly whether the same exact recovery and the subsequent dimension-free bound continue to hold for non-Gaussian noising schedules or only under the stated Gaussian assumption.
[§5] §5 (experiments): the reported outperformance is presented without error bars, dataset sizes, or details on how the conditional mutual information was estimated or controlled; these omissions make it difficult to assess whether the observed gains are statistically robust or sensitive to hyper-parameter choices that implicitly affect the initialization error term.

minor comments (2)

Notation for the tangent and normal components should be introduced once with a clear diagram or equation reference rather than being redefined in multiple sections.
The abstract claims 'dimension-free' bounds; the manuscript should confirm that the constant hidden in the O(·) notation is indeed independent of ambient dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (normal-tangent decomposition and Theorem 1): the proof that the observed-direction score is exactly recovered from the linear measurement appears to rely on the forward process being exactly Gaussian; the manuscript should state explicitly whether the same exact recovery and the subsequent dimension-free bound continue to hold for non-Gaussian noising schedules or only under the stated Gaussian assumption.

Authors: The analysis in §3 is developed under the Gaussian noising assumption, which is stated explicitly in the abstract and at the opening of §3. The exact recovery of the observed-direction score follows directly from the Gaussian property of the forward process (specifically, the fact that the conditional distribution remains Gaussian). For non-Gaussian noising schedules the exact recovery does not hold in general, and the subsequent dimension-free bound on the tangent-score error would require additional assumptions on the noise distribution. We will add a clarifying remark in §3 that makes the scope of the result explicit and notes that the Gaussian assumption is essential for the exact recovery step. revision: yes
Referee: [§5] §5 (experiments): the reported outperformance is presented without error bars, dataset sizes, or details on how the conditional mutual information was estimated or controlled; these omissions make it difficult to assess whether the observed gains are statistically robust or sensitive to hyper-parameter choices that implicitly affect the initialization error term.

Authors: We agree that the experimental presentation would be strengthened by these details. In the revised manuscript we will report error bars computed over multiple independent runs, specify the exact dataset sizes used for each task, and add a paragraph describing how the conditional mutual information is estimated (including the approximation method and any controls employed). We will also briefly discuss sensitivity of the observed gains to hyper-parameters that influence the initialization error term. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivation establishes an information-theoretic upper bound on tangent-component score mismatch error by conditional mutual information between observed and unobserved components. This bound and the subsequent decomposition into initialization plus pathwise errors follow directly from the stated Gaussian noising assumption and the normal-tangent decomposition of the score; neither step reduces to a fitted parameter, a self-citation chain, or a redefinition of inputs. The resulting guarantees are therefore self-contained against external benchmarks once the forward process is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard diffusion-model assumptions rather than new free parameters or invented entities.

axioms (2)

domain assumption The forward noising process is Gaussian.
Required for the claim that the observed-direction score is exactly determined by the measurement.
domain assumption A pretrained unconditional diffusion model is available.
Enables the zero-shot conditional sampling setting.

pith-pipeline@v0.9.0 · 5501 in / 1352 out tokens · 60853 ms · 2026-05-08T17:41:03.307759+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 / Foundation.GeneralizedDAlembert washburn_uniqueness_aczel / dAlembert_cosh_solution_aczel unclear

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

We analyze this issue through a normal–tangent decomposition of the score function. For Gaussian noising, the observed-direction score is exactly determined by the measurement; only the tangent conditional score is unknown.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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