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arxiv: 2606.28417 · v2 · pith:3BO5QL7Qnew · submitted 2026-06-25 · 💻 cs.CV

DiffRGD: An Inference-Time Diffusion Guidance Through Riemannian Gradient Descent

Jia-Wei Liao , Li-Xuan Peng , Mei-Heng Yueh , Min Sun , Cheng-Fu Chou , Jun-Cheng Chen This is my paper

Pith reviewed 2026-07-03 22:49 UTC · model grok-4.3

classification 💻 cs.CV
keywords diffusion modelsinference-time guidanceRiemannian gradient descentspherical manifoldimage restorationconditional generationgenerative modelingdistribution preservation
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The pith

Diffusion guidance can steer samples while exactly preserving the Gaussian latent distribution by optimizing on a spherical manifold.

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

Diffusion models generate by reversing a noise process whose latents stay Gaussian at each step. Standard inference-time guidance uses a differentiable objective to steer toward desired outputs, but this usually shifts the latent distribution and lowers sample quality. DiffRGD instead treats every sampling step as a constrained optimization problem whose feasible set is the sphere induced by the current Gaussian variance, then solves the problem with Riemannian gradient descent. The resulting updates keep the latent on the manifold required by the diffusion process. The method is plug-and-play for any pre-trained model and improves results on image restoration and conditional generation tasks.

Core claim

DiffRGD formulates each sampling step as a constrained optimization problem on a spherical manifold induced by the latent Gaussian distribution, and solves it efficiently via Riemannian Gradient Descent (RGD) to explicitly preserve the latent Gaussian structure.

What carries the argument

Riemannian gradient descent on the spherical manifold induced by the latent Gaussian distribution at each diffusion step

If this is right

  • The method integrates directly into any pre-trained diffusion model without retraining or fine-tuning.
  • It reduces the distributional drift introduced by conventional guidance techniques.
  • It improves performance over prior inference-time guidance methods on image restoration and conditional generation tasks.
  • The manifold constraint keeps each sampling trajectory consistent with the diffusion model's original Gaussian assumptions.

Where Pith is reading between the lines

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

  • The same manifold-constrained optimization idea could apply to other iterative samplers that rely on a fixed distributional form at each step.
  • Respecting the geometry of the latent space during guidance may generalize to non-image domains such as audio or 3D shape generation.
  • If the Gaussian assumption is relaxed, analogous constraints on other manifolds could be derived for models with different latent distributions.

Load-bearing premise

The latent at each diffusion step remains exactly Gaussian so that the spherical manifold accurately captures the allowable set.

What would settle it

Measure the empirical variance and moments of the guided latents after several sampling steps and compare them to the expected Gaussian; large deviation would falsify the preservation claim.

Figures

Figures reproduced from arXiv: 2606.28417 by Cheng-Fu Chou, Jia-Wei Liao, Jun-Cheng Chen, Li-Xuan Peng, Mei-Heng Yueh, Min Sun.

Figure 1
Figure 1. Figure 1: Illustration of our method: Previ￾ous works perform guidance without latent￾distribution-aware constraints, which can re￾sult in the off-manifold phenomena that hinder the sample quality; DiffRGD pro￾poses a latent-distribution-aware geometry constraint tailored to the Gaussian properties of the diffusion model. However, their guidance operates in the ambient space that cannot preserve the stepwise Gaussia… view at source ↗
Figure 2
Figure 2. Figure 2: Gaussian distribution-aware spheri￾cal manifold construction. The insight of Proposition 1 is that we aim to construct the spherical manifold based on the same density as the original Gaussian of x t . The radius σt r follows a scaled￾chi distribution centered close to p nσt with two-sided probability decreasing. We show the conceptual illustra￾tion in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of DiffRGD [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison for image restoration tasks on FFHQ 256 × 256 validation set. Our proposed method achieves better restoration results with fewer artifacts than other compared methods. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative results of super-resolution by our DiffRGD method with Stable Diffusion under ×8 and ×12 subsampling rates. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results for conditional generation tasks on CelebA-HQ 256 × 256 validation set. (a) seg￾mentation maps (b) sketches (c) FaceID. Our method generates results that better align with the given conditions than other state-of-the-art methods. Appendix. DiffRGD shows stable performance across guidance strengths, whereas DSG performs poorly across all settings [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results for conditional generation tasks on CelebA-HQ 256 × 256 validation set. (a) seg￾mentation maps (b) sketches (c) FaceID. Our method generates results that better align with the given conditions than other state-of-the-art methods. 4.3 Conditional Generation We further evaluate our method against representative baselines on three conditional generation tasks: segmentation maps, sketches, … view at source ↗
Figure 7
Figure 7. Figure 7: illustrates the relationship over the number of inner iterations and the norm of the Riemannian gradient at different timesteps. While Theorem 1 guarantees a convergence rate of O (1/ p k), the figure shows that the norm of the Riemannian gradient already becomes sufficiently small when K = 3 in practice. This supports our experiment settings in [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of different guidance strengths for segmentation map-guided conditional generation [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of different guidance strengths for segmentation map-guided conditional generation. E Additional Results We present additional qualitative results for image restoration in [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: illustrates the relationship over the number of inner iterations and the norm of the Riemannian gradient at different timesteps. While Theorem 1 guarantees a convergence rate of O (1/ p k), the figure shows that the norm of the Riemannian gradient already becomes sufficiently small when K = 3 in practice. This supports our experiment settings in [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative comparisons for image restoration tasks on FFHQ 256 × 256 validation set. Please refer to the highlighted regions for detailed comparison. Guidance FreeDoM DSG ADMMDiff DiffRGD (Ours) Segmentation Map Sketch FaceID 3 [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative comparisons for image restoration tasks on FFHQ 256 × 256 validation set. Please refer to the highlighted regions for detailed comparison. Guidance FreeDoM DSG ADMMDiff DiffRGD (Ours) Segmentation Map Sketch FaceID 3 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Qualitative comparison for conditional generation tasks on CelebA-HQ 256 × 256 validation set. (a) segmentation maps to human faces (b) sketches to human faces (c) FaceID to human faces. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Qualitative comparison for conditional generation tasks on CelebA-HQ 256 × 256 validation set. (a) segmentation maps to human faces (b) sketches to human faces (c) FaceID to human faces. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Super-resolution. Ground Truth DPS DSG ADMMDiff DiffRGD (Ours) 1000 steps 100 steps MPGD Measurement [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: Qualitative comparisons on different DDIM timesteps: Compared to 1000 timesteps, our method can perform well even in 100 steps, please refer to the highlighted regions for detailed comparison. Although the quality is close to ADMMDiff, our method demands less time. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Gaussian deblurring [PITH_FULL_IMAGE:figures/full_fig_p034_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Qualitative comparison for image denoising tasks on ImageNet 256 × 256. Please refer to the high￾lighted regions for detailed comparison. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: Qualitative comparison on the style-guided generation task using the Stable Diffusion v1.5 at a reso￾lution of 512 × 512. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Qualitative comparison on the style-guided generation task using the Stable Diffusion v1.5 at a reso￾lution of 512 × 512. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
read the original abstract

Recently, diffusion models have been widely adopted in generative modeling and have served as foundational models for many image generation tasks. To control the generation without costly re-training or fine-tuning, many works seek inference-time guidance methods to steer the latent via a differentiable objective at inference time. However, these methods cannot effectively preserve the original Gaussian distribution because they introduce distributional drift, thereby degrading the sample quality. To address this gap, we propose DiffRGD, a distribution-aware guidance framework that explicitly preserves the latent Gaussian structure. DiffRGD formulates each sampling step as a constrained optimization problem on a spherical manifold induced by the latent Gaussian distribution, and solves it efficiently via Riemannian Gradient Descent (RGD). DiffRGD is a plug-and-play method that can be seamlessly integrated into any pre-trained diffusion model. Extensive experiments demonstrate that DiffRGD outperforms previous methods in most image restoration and conditional generation tasks. Our project page is available at https://diffrgd.github.io/.

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 / 2 minor

Summary. The paper proposes DiffRGD, an inference-time guidance method for pre-trained diffusion models. It formulates each reverse sampling step as a constrained optimization problem on the spherical manifold induced by the latent Gaussian N(0, σ_t^{2}I), solved via Riemannian gradient descent to explicitly avoid distributional drift while steering generation according to a differentiable objective. The method is presented as plug-and-play and is reported to outperform prior guidance techniques on image restoration and conditional generation tasks.

Significance. If the distribution-preservation property holds, the approach would address a recognized limitation of existing inference-time guidance methods by keeping the latent on the correct manifold at each step, which could improve sample quality without retraining. The plug-and-play design is a practical advantage for adoption with existing models.

major comments (2)
  1. [Method section (formulation of constrained optimization and RGD update)] The central claim that the RGD update on the sphere 'explicitly preserves the latent Gaussian structure' is load-bearing yet unsupported by any derivation showing that the projected step leaves the marginal invariant under the true reverse kernel when the incoming latent deviates from exact Gaussianity (due to prior guidance, model error, or discretization). No such invariance proof or error analysis appears in the method description.
  2. [Method and Experiments] The assumption that the allowable set is exactly the sphere induced by N(0, σ_t^{2}I) at every guided step is used without justification or sensitivity analysis; any deviation makes the manifold constraint incorrect, yet the paper supplies no experiment isolating this effect (e.g., measuring KL divergence to the unguided marginal after guidance).
minor comments (2)
  1. [Abstract] The abstract states that DiffRGD 'outperforms previous methods in most image restoration and conditional generation tasks' but provides no quantitative metrics, datasets, or baselines in the summary; these should be stated explicitly.
  2. [Method] Notation for the spherical constraint and the precise form of the Riemannian gradient step should be introduced with an equation number for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Method section (formulation of constrained optimization and RGD update)] The central claim that the RGD update on the sphere 'explicitly preserves the latent Gaussian structure' is load-bearing yet unsupported by any derivation showing that the projected step leaves the marginal invariant under the true reverse kernel when the incoming latent deviates from exact Gaussianity (due to prior guidance, model error, or discretization). No such invariance proof or error analysis appears in the method description.

    Authors: We acknowledge that the manuscript does not contain a formal derivation establishing invariance of the marginal under the true reverse kernel when the incoming latent deviates from exact Gaussianity. The RGD step is constructed to enforce the spherical constraint induced by the known variance schedule at each timestep, which prevents the latent from leaving the manifold on which the diffusion process is defined. We agree that an explicit error analysis or invariance argument under deviations would strengthen the theoretical grounding. In the revised manuscript we will add a dedicated paragraph in the method section discussing the approximation, its relation to the reverse kernel, and any available bounds on the introduced error. revision: yes

  2. Referee: [Method and Experiments] The assumption that the allowable set is exactly the sphere induced by N(0, σ_t^{2}I) at every guided step is used without justification or sensitivity analysis; any deviation makes the manifold constraint incorrect, yet the paper supplies no experiment isolating this effect (e.g., measuring KL divergence to the unguided marginal after guidance).

    Authors: The spherical constraint follows directly from the forward process definition: at timestep t the marginal is N(0, σ_t²I), whose high-dimensional mass concentrates on the sphere of radius σ_t. This is stated in Section 3.1. We do not currently report an explicit KL-divergence measurement between guided and unguided marginals. Our experiments instead evaluate downstream sample quality and guidance fidelity. We will add a sensitivity study (including KL estimates on a subset of timesteps) to the experimental section of the revision to quantify how closely the guided latents remain to the unguided marginal. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The provided abstract and description present DiffRGD as a novel formulation of each diffusion sampling step as a constrained Riemannian optimization on the sphere induced by the latent Gaussian. No equations, fitted parameters, self-citations, or prior results are shown that would reduce this claim to a definition, a renamed input, or a self-referential fit. The method is described as plug-and-play integration into pre-trained models, with the Gaussian-preservation property asserted as a direct consequence of the manifold constraint rather than derived from any internal tautology or load-bearing self-citation. This is the common case of an honest non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that diffusion latents remain Gaussian at every step (inducing the spherical constraint) and on the unverified effectiveness of RGD in eliminating drift; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Latent variables remain exactly Gaussian at each sampling step, inducing a spherical manifold on which guidance must be performed.
    This premise directly defines the constrained optimization problem described in the abstract.

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