Noise is All You Need: Solving Linear Inverse Problems by Noise Combination Sampling with Diffusion Models

Hiroyuki Kasai; Xun Su

arxiv: 2510.23633 · v2 · submitted 2025-10-24 · 💻 cs.LG · cs.AI· cs.CV· eess.IV

Noise is All You Need: Solving Linear Inverse Problems by Noise Combination Sampling with Diffusion Models

Xun Su , Hiroyuki Kasai This is my paper

Pith reviewed 2026-05-18 04:32 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CVeess.IV

keywords diffusion modelslinear inverse problemsnoise combination samplingmeasurement scoreDDPM samplingzero-shot solvingimage compressiongenerative models

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The pith

Synthesizing an optimal noise vector from a subspace lets diffusion models embed measurement constraints directly into generation without per-step tuning.

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

The paper seeks to fix a core dilemma in zero-shot inverse problem solving with pretrained diffusion models: too much observation data breaks the generative process while too little fails to enforce the inverse constraints. It introduces Noise Combination Sampling to build a single noise vector from a noise subspace that approximates the measurement score and replaces the usual noise term inside the standard DDPM update. This substitution embeds the conditional information naturally at every step. A reader would care because the change requires no extra hyperparameter schedules, adds almost no cost, and yields more stable results especially when only a few generation steps are used, covering tasks such as image compression.

Core claim

We propose Noise Combination Sampling, a novel method that synthesizes an optimal noise vector from a noise subspace to approximate the measurement score, replacing the noise term in the standard Denoising Diffusion Probabilistic Models process. This enables conditional information to be naturally embedded into the generation process without reliance on step-wise hyperparameter tuning. Our method can be applied to a wide range of inverse problem solvers, including image compression, and, particularly when the number of generation steps T is small, achieves superior performance with negligible computational overhead, significantly improving robustness and stability.

What carries the argument

Noise Combination Sampling: synthesizes an optimal noise vector from a noise subspace to approximate the measurement score and substitutes it for the standard noise term inside the DDPM sampling step.

If this is right

The approach applies to a wide range of inverse-problem solvers that already use pretrained diffusion models.
Superior performance appears particularly when the number of generation steps T is small.
Computational overhead remains negligible compared with existing conditional diffusion solvers.
Robustness and stability of the overall generation process increase without manual tuning schedules.

Where Pith is reading between the lines

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

The same noise-subspace idea could be tested on non-linear inverse problems where the measurement operator is not a simple matrix.
If the subspace construction generalizes, practitioners might stop training separate conditional diffusion models for each new inverse task.
Exploring how the dimension or selection of the noise subspace affects approximation error would give a practical knob for different measurement types.

Load-bearing premise

An optimal noise vector synthesized from a noise subspace can accurately approximate the measurement score while preserving the integrity of the pretrained diffusion generative process.

What would settle it

Running the same inverse-problem benchmarks with and without Noise Combination Sampling and checking whether the synthesized-noise version produces outputs that satisfy the linear measurements more closely than standard diffusion sampling, especially at small T, would confirm or refute the central claim.

read the original abstract

Pretrained diffusion models have demonstrated strong capabilities in zero-shot inverse problem solving by incorporating observation information into the generation process of the diffusion models. However, this presents an inherent dilemma: excessive integration can disrupt the generative process, while insufficient integration fails to emphasize the constraints imposed by the inverse problem. To address this, we propose \emph{Noise Combination Sampling}, a novel method that synthesizes an optimal noise vector from a noise subspace to approximate the measurement score, replacing the noise term in the standard Denoising Diffusion Probabilistic Models process. This enables conditional information to be naturally embedded into the generation process without reliance on step-wise hyperparameter tuning. Our method can be applied to a wide range of inverse problem solvers, including image compression, and, particularly when the number of generation steps $T$ is small, achieves superior performance with negligible computational overhead, significantly improving robustness and stability.

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's main move is swapping standard DDPM noise for a synthesized vector from a noise subspace to approximate the measurement score and skip per-step tuning in inverse problem solvers.

read the letter

The colleague should know that this paper introduces Noise Combination Sampling for diffusion models on linear inverse problems. It builds an optimal noise vector from a subspace to stand in for the measurement score inside the standard DDPM update, so conditional information gets embedded without having to tune guidance strength at every step. The claim is that this works especially well at small generation step counts T and adds almost no cost, with applications to image compression and similar tasks. That framing targets a real usability pain point in existing diffusion inverse solvers. What the paper does reasonably well is identify the over- or under-integration dilemma and offer a direct fix that avoids extra hyperparameters. If the experiments later show stable gains over tuned baselines on standard imaging benchmarks, with visuals or metrics that hold across different T, the practical payoff is clear. The method is simple enough that it could be tried quickly on top of existing solvers. The soft spot is the substitution itself. Standard DDPM noise is tied to the trained variance schedule and supplies the exact stochasticity the model expects. Replacing it with a custom vector chosen to match a measurement gradient risks shifting the effective mean or variance at each step, particularly when T is small. The abstract gives no derivation showing the marginals stay close or any error bound, so the stability and robustness claims rest on whether the experiments actually check for distribution shift or sample quality drop. If those checks are missing or weak, the central approximation needs more support. This is for people already working on zero-shot diffusion solvers for inverse problems in machine learning, such as imaging or compression pipelines. A reader who wants a tuning-free option and is willing to verify the noise swap empirically would get the most out of it. I would send it to peer review. The idea is concrete, the target problem is practical, and the work is coherent enough on its own terms to deserve referee time even if the theory needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Noise Combination Sampling for zero-shot linear inverse problem solving with pretrained diffusion models. It synthesizes an optimal noise vector from a noise subspace to approximate the measurement score, then substitutes this vector for the standard noise term in the DDPM reverse process. The approach is claimed to embed conditional information naturally without step-wise hyperparameter tuning, yielding superior performance and stability especially at small T with negligible overhead, and to be compatible with existing solvers such as those for image compression.

Significance. If the substitution preserves the pretrained generative trajectory, the method would offer a practical route to stable, tuning-free integration of measurements into diffusion sampling. This could be particularly useful in low-step regimes where existing guidance-based or optimization-based solvers become unstable.

major comments (2)

[Method] The central substitution step (described in the method section) replaces the calibrated Gaussian noise term whose variance schedule is fixed by the pretrained DDPM training. No derivation is supplied showing that the synthesized vector from the noise subspace leaves the marginal distribution p(x_t) or the stochastic trajectory unchanged; without an error bound or invariance argument, the claim that the pretrained generative process remains intact is unsupported and load-bearing for all stability assertions.
[Experiments] When T is small the accumulated approximation error from using a non-standard noise vector at each step could shift the effective mean or variance; the experiments section should include an ablation that isolates this effect (e.g., by comparing trajectories with and without the substitution while holding the measurement approximation fixed).

minor comments (2)

Clarify the precise construction of the noise subspace and the optimization criterion used to select the optimal vector; the current description leaves the procedure under-specified for reproduction.
Add a short complexity analysis or wall-clock timing table to substantiate the 'negligible computational overhead' claim relative to the baseline solvers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and agree that both points warrant revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Method] The central substitution step (described in the method section) replaces the calibrated Gaussian noise term whose variance schedule is fixed by the pretrained DDPM training. No derivation is supplied showing that the synthesized vector from the noise subspace leaves the marginal distribution p(x_t) or the stochastic trajectory unchanged; without an error bound or invariance argument, the claim that the pretrained generative process remains intact is unsupported and load-bearing for all stability assertions.

Authors: We agree that the manuscript lacks a formal derivation or error bound establishing invariance of p(x_t) or the stochastic trajectory under the substitution. In the revision we will add a theoretical subsection deriving a bound on the deviation induced by the synthesized noise vector, leveraging the subspace construction and the fact that the approximation targets the measurement score while remaining within the noise manifold of the pretrained model. This will directly support the stability claims. revision: yes
Referee: [Experiments] When T is small the accumulated approximation error from using a non-standard noise vector at each step could shift the effective mean or variance; the experiments section should include an ablation that isolates this effect (e.g., by comparing trajectories with and without the substitution while holding the measurement approximation fixed).

Authors: We agree that an ablation isolating the substitution effect is necessary to quantify any accumulated shifts for small T. In the revised experiments we will add a controlled comparison of trajectories with and without the noise substitution (measurement approximation held fixed) and report statistics on mean/variance deviation across steps, with particular emphasis on low-T regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: method introduces new sampling procedure without reducing to fitted inputs or self-citations by construction

full rationale

The provided abstract and description outline a proposed Noise Combination Sampling technique that synthesizes a noise vector from a subspace to approximate the measurement score in DDPM. No equations, fitting procedures, or derivation steps are exhibited that equate a claimed prediction or result directly to its own inputs or prior self-citations. The central premise relies on the substitution preserving the generative process, but this is presented as a novel construction rather than a tautological renaming or load-bearing self-reference. The derivation chain appears self-contained as an empirical proposal without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are described in sufficient detail to populate the ledger.

pith-pipeline@v0.9.0 · 5685 in / 1058 out tokens · 37659 ms · 2026-05-18T04:32:05.630358+00:00 · methodology

discussion (0)

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