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arxiv: 2606.28785 · v1 · pith:27ZPGMLSnew · submitted 2026-06-27 · 💻 cs.CV

Stochastic Optimal Control Sampling for Diffusion Inverse Problems

Jie Zhang , Youmei Qiu , Hanling Tian , Jingyuan Zhang , Xiang Yin , Xiaolin Huang This is my paper

Pith reviewed 2026-06-30 09:43 UTC · model grok-4.3

classification 💻 cs.CV
keywords diffusion modelsinverse problemsstochastic optimal controlimage reconstructionsampling methodsdenoisinglinear SDE
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The pith

Stochastic optimal control sampling derives a closed-form update applied at each diffusion denoising step to steer trajectories toward measurements.

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

The paper establishes that diffusion models can solve image inverse problems by treating the denoising process as a dynamical system and injecting control via stochastic optimal control. It shows that a closed-form control update can be computed and applied independently at every sampling step, guiding the trajectory to satisfy measurements without optimizing the full path. This per-step approach replaces expensive trajectory-wide optimization used in earlier SOC methods. Control strength can be modulated to better match the diffusion model's capabilities, yielding improved perceptual quality in the output. The technique remains compatible with multiple linear stochastic differential equation backbones and is tested on a range of inverse tasks.

Core claim

SOCS models the denoising process as a dynamical system and injects control signals via SOC. Previous SOC-based approaches address inverse problems by optimizing over the entire trajectory, which is computationally expensive. In contrast, SOCS derives a closed-form control update and applies it at each sampling step, pulling the measurement-consistent clean prediction back onto the denoising flow. In SOCS, the control strength can be modulated to align with the diffusion model's native capabilities and thereby enhance perceptual quality. The method is compatible with a variety of linear stochastic differential equation backbones.

What carries the argument

The closed-form control update derived from stochastic optimal control, applied independently at each sampling step of the linear SDE diffusion process.

Load-bearing premise

The closed-form control update remains valid when applied independently at each sampling step without requiring re-optimization of the full trajectory or violating the underlying linear SDE assumptions of the diffusion backbone.

What would settle it

A direct comparison showing that repeated per-step application of the closed-form update produces trajectories whose measurement consistency or sample quality deviates substantially from the optimum obtained by full-trajectory SOC optimization.

Figures

Figures reproduced from arXiv: 2606.28785 by Hanling Tian, Jie Zhang, Jingyuan Zhang, Xiang Yin, Xiaolin Huang, Youmei Qiu.

Figure 1
Figure 1. Figure 1: Illustration of our method (an example on a random inpainting task). [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative samples of SOCS. Our method leverages stochastic optimal control (SOC) to provide an efficient, theoretically grounded framework for diffusion inverse problems. In (a)(b), we present VP-SDE results on FFHQ and ImageNet-256, respectively; in (c)(d), we showcase VE-SDE results on LSUN Church and FFHQ; in (e)(f), we show natural images at a resolution of 512. flow-matching models (represented b… view at source ↗
Figure 3
Figure 3. Figure 3: SOCS vs DAPS on 2D synthetic data. SOCS exhibits a more direct refinement trajectory [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sample diversity. We present several diverse samples generated by the SOCS under two sparse measurements. SOCS produces a variety of samples with distinct features, including differences in expression, wearings, and hairstyles. with the qualitative comparisons in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 22
Figure 22. Figure 22: Detailed prompts, theoretical derivations, and implementation specifics [PITH_FULL_IMAGE:figures/full_fig_p013_22.png] view at source ↗
Figure 6
Figure 6. Figure 6: Quantitative evaluation of image quality metrics as a function of the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: SOCS, DAPS, and DPS (both SDE and ODE variants) on two-dimensional synthetic data. We further design a simple nonlinear measurement model y = exp − ∥x∥ 2 0.5  + exp − ∥x−(0.5,0.5)∥ 2 0.5  +n, where n\sim \mathcal {N}(0,\beta _y^2) with \beta _y=0.3 , to match our inverse problem setting. When y=1.50 , the prior is bimodal, yet the likelihood induced by the nonlinear measurement places most of its mass … view at source ↗
Figure 9
Figure 9. Figure 9: SOCS vs. prior-gradient guidance in Langevin refinement. SOCS follows a straighter, lower￾energy path to reach data consistency. which is consistent with the stochastic optimal control principle of achieving the target with minimal control energy. In addition, we investigate the difference between using the standard mea￾surement gradient ∇xt ∥H(xt)−y∥ 2 2 in the Langevin refinement (Eq. (8)) and us￾ing the… view at source ↗
Figure 10
Figure 10. Figure 10: Sampling results of SOCS-SD1.5 on FFHQ 256 × 256 images. The sampling is enhanced with classifier-free guidance for text with guidance scale 7.5. The used text prompt is "a natural looking human face". Let E : R n → R k and D : R k → R n denote the encoder and decoder of the LDM, respectively. We extend Eq. 6 to the latent space and perform the corresponding optimization over latent variables: \begingroup… view at source ↗
Figure 11
Figure 11. Figure 11: Quantitative evaluations of image quality for different numbers of [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qualitative samples across different numbers of function evaluations [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Quantitative evaluations of image quality for different ODE steps. [PITH_FULL_IMAGE:figures/full_fig_p041_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Qualitative samples across different ODE steps. [PITH_FULL_IMAGE:figures/full_fig_p041_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Quantitative evaluations of image quality for different denoising [PITH_FULL_IMAGE:figures/full_fig_p041_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Image quality consistently improves with increasing p quantitatively in [PITH_FULL_IMAGE:figures/full_fig_p042_16.png] view at source ↗
Figure 20
Figure 20. Figure 20: We present additional results for SD v1.5 in [PITH_FULL_IMAGE:figures/full_fig_p042_20.png] view at source ↗
Figure 17
Figure 17. Figure 17: Qualitative demonstration of the impact of [PITH_FULL_IMAGE:figures/full_fig_p043_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Phase retrieval samples from SOCS across four independent runs [PITH_FULL_IMAGE:figures/full_fig_p044_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Denoising processes for all tasks with VP-SDE [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 19
Figure 19. Figure 19: Denoising processes for all tasks (continued) [PITH_FULL_IMAGE:figures/full_fig_p045_19.png] view at source ↗
Figure 19
Figure 19. Figure 19: Denoising processes for all tasks (continued) [PITH_FULL_IMAGE:figures/full_fig_p046_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Denoising processes for Inpaint tasks with VE-SDE [PITH_FULL_IMAGE:figures/full_fig_p047_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Denoising processes for SD v1.5 on natural images. [PITH_FULL_IMAGE:figures/full_fig_p048_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Denoising processes for SD3-medium on FFHQ images. [PITH_FULL_IMAGE:figures/full_fig_p049_22.png] view at source ↗
read the original abstract

Benefiting from the strong ability to capture data distributions, diffusion models have become powerful tools for solving image inverse problems. The key is to controllably steer the sampling trajectory toward the measurements while respecting the diffusion prior. In this work, we introduce Stochastic Optimal Control Sampling (SOCS), which models the denoising process as a dynamical system and injects control signals via SOC. Previous SOC-based approach addresses inverse problems by optimizing over the entire trajectory, which is computationally expensive. In contrast, we derive a closed-form control update and apply it at each sampling step, pulling the measurement-consistent clean prediction back onto the denoising flow. In SOCS, we can readily modulate the control strength to align with the diffusion model's native capabilities and thereby enhance perceptual quality. Our method is compatible with a variety of linear stochastic differential equation backbones. Extensive experiments across a broad spectrum of image inverse tasks demonstrate that SOCS achieves accurate measurement-aligned reconstructions with improved visual fidelity and stronger quantitative performance.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces Stochastic Optimal Control Sampling (SOCS) for diffusion-based image inverse problems. It models the denoising process as a dynamical system, derives a closed-form control update applied independently at each sampling step to pull measurement-consistent predictions onto the denoising flow, and claims this is more efficient than full-trajectory optimization while remaining compatible with linear SDE backbones. Control strength can be modulated to improve perceptual quality, with experiments across inverse tasks showing better measurement alignment and visual fidelity.

Significance. If the closed-form per-step derivation holds without violating SOC optimality or SDE linearity, the method would offer a computationally lighter alternative to trajectory-wide optimization for steering diffusion models in inverse problems, with explicit control over the fidelity-perception trade-off.

major comments (2)
  1. Abstract: the claim of a closed-form control update that solves the underlying SOC problem when applied independently at each denoising step (without full-horizon re-optimization) is presented without any equations, value-function derivation, or proof; this is load-bearing for the efficiency and correctness assertions and cannot be assessed from the given text.
  2. Abstract: the statement that the update 'pulls the measurement-consistent clean prediction back onto the denoising flow' leaves open whether the instantaneous correction preserves the linear SDE assumptions or implicitly sets future controls to zero, which would make the trajectory deviate from the true SOC optimum as noted in the stress-test concern.
minor comments (1)
  1. The abstract asserts 'extensive experiments' and 'stronger quantitative performance' but provides no task list, metrics, or baseline comparisons in the visible text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for highlighting these points about the abstract. We address each major comment below.

read point-by-point responses

  1. Referee: Abstract: the claim of a closed-form control update that solves the underlying SOC problem when applied independently at each denoising step (without full-horizon re-optimization) is presented without any equations, value-function derivation, or proof; this is load-bearing for the efficiency and correctness assertions and cannot be assessed from the given text.

    Authors: The abstract is a concise summary. The complete derivation of the closed-form per-step control—including the value-function formulation, the optimality conditions under the linear SDE, and the justification that independent application at each step solves the SOC problem without full-horizon re-optimization—is given in Section 3 of the manuscript. This material directly supports the efficiency and correctness claims. We are willing to add a parenthetical reference to the key result in the abstract if the editor considers it helpful. revision: partial

  2. Referee: Abstract: the statement that the update 'pulls the measurement-consistent clean prediction back onto the denoising flow' leaves open whether the instantaneous correction preserves the linear SDE assumptions or implicitly sets future controls to zero, which would make the trajectory deviate from the true SOC optimum as noted in the stress-test concern.

    Authors: The derivation shows that the control is recomputed at every step from the current state; it does not implicitly set future controls to zero. The instantaneous correction is constructed to keep the trajectory on the linear SDE flow while satisfying the measurement constraint at that instant, and the overall trajectory remains consistent with the SOC optimum. This is further supported by the stress-test experiments reported in the paper. revision: no

Circularity Check

0 steps flagged

No circularity detected in derivation of closed-form SOC update

full rationale

The paper presents a derivation of a closed-form control update for applying stochastic optimal control at each denoising step independently. No equations or claims in the provided abstract or description reduce the result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The approach is framed as a general mathematical derivation compatible with linear SDE backbones, with the central claim resting on standard SOC principles rather than tautological re-use of the target quantities. This is the expected non-finding for a derivation-focused methods paper whose assumptions are stated externally to the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the method description is too high-level to enumerate any.

pith-pipeline@v0.9.1-grok · 5699 in / 979 out tokens · 54107 ms · 2026-06-30T09:43:25.419007+00:00 · methodology

discussion (0)

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    To interpret SOCS in terms of measure change and posterior sampling, we consider the stochastic counterpart with the same drift/control injection: dxt = (ftxt +g tut,γ) dt+g t dWt, x 0 ∼µ 0, (54) and the native diffusion obtained by settingut,γ ≡0. Denote the induced path measures onx0:T byQ path andP path, and the terminal marginals byQT andP T. By Eq. (...

  52. [52]

    a natural looking human face

    We further set the total number of Langevin steps toN= 50, consistent with the default setting in [47]. – FlowChefWe adopt the protocol in [19], which selects hyperparameters via a grid search on 100 images. Accordingly, we set the step size to 200 for super-resolution tasks and 50 for deblurring tasks. – FlowDPSFor data-consistency optimization, we follo...

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