Kuramoto Orientation Diffusion Models

We appreciate $\textcolor{blue}{\text{Reviewer ZpUX}}$’s recognition of our method’s novelty and thank you for the thorough feedback and constructive suggestions. Below, we address each of their points in detail:

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1. Reducing the O(T) Simulation Cost

Although our local score matching requires sample pairs $(\theta_t,\theta_{t+1})$ along the forward chain, this cost actually can be nearly eliminated with precomputation:

Pre‐simulate & cache: Before training, we can run the forward SDE on the entire dataset, save all $(\theta_t,\theta_{t+1})$ pairs to disk, and then load them directly during training. This makes the simulation cost effectively O(0) at each training step.

Epoch‐wise precompute: Otherwise, if disk space is limited, we can simply re‐generate and cache one epoch’s worth of pairs at the start of each epoch. This still dramatically reduces the simulation overhead.

We will detail the above discussion of reducing training costs in the limitations section.

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2. Intuition for General vs. Orientation‑Rich Images

Our Kuramoto coupling is foremost a synchronization mechanism: it excels at pulling similar phases together, thereby preserving edges and repetitive patterns (see qualitative example in Fig. 1 and the SNR over time in Fig. 5). On orientation‑rich data (fingerprints, textures) where the data are dominated by the simple outlines and repeating ridge structures, this yields sharp samples in few steps.

By contrast, natural images (e.g., CIFAR‑10) demand modeling complex, global semantics (object shapes, color gradients, backgrounds), often characterized by higher order longer-range correlations. In this setting, the local synchronization bias becomes less relevant and potentially even detrimental; and an isotropic diffusion process with a global drift may be better suited to capture these large‑scale, non‑repetitive features. As a result, our method trades some global fidelity (reflected in higher CIFAR‑10 FID) in exchange for strong local coherence.

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3. Comparisons: Blurring Diffusion & Reference-only Process

We thank the reviewer for the constructive suggestion of comparing with blurring diffusion models. Below, we compare against blurring diffusion models [16,35] with linear isotropic drifts, and a reference-only variant (setting $K(t)=0$) on Brodatz textures, to isolate the benefit of nonlinear, non‑isotropic coupling:

FID on Brodatz Textures

Reference-only Process. Setting $K(t)=0$ yields an isotropic drift toward a single reference phase, completely ignoring any neighbor information. Also, even with only the isotropic reference drift, the marginal transitions are still intrackable due to the phase wrapping, forcing us to rely on the local score matching trick again. The FID worsens by around 15 points at 100 steps,

Blurring Diffusion Models. Although these two models [35,16] exhibit blurring behavior, they actually apply linear isotropic drift in the SDE. The spatially uniform drift cannot selectively preserve orientation patterns. Therefore, they fall far short of our model at 100 steps.

Together, these comparisons demonstrate that nonlinear, non-isotropic phase coupling is the key ingredient for rapid, structure‐preserving diffusion on orientation‑rich data.

In the Related Work section, we will also cite [C], which develops spherical von Mises–Fisher diffusion on $S^d$, which generalizes the circular von Mises we use on $S^1$.

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4. Longer Sampling Chains

We agree that more sampling steps can indeed further refine results. In preliminary runs on Brodatz:

FID at 1,000 steps: 14.19

FID at 2,000 steps: 13.52

FID at 4,000 steps: 12.89

We see continued improvement as the denoising step increases. This suggests that in the CIFAR‑10 case, extending to 4,000 steps may likewise narrow the FID gap.

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5. Corrections of $\nabla \log p(\theta)$

Thanks for catching the typo. The derivation of Eq. (5) is correct, but line 38 is indeed wrong. We have corrected it in the revision.

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6. Additional Experiments on High‑Resolution & Vector‑Valued Data

To further validate our method’s generality, we evaluated our model on two new datasets:

High-resolution Ground Terrain [A] (3x128x128). Compared with Brodatz textures, this dataset covers diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules:

FID on Ground Terrain Dataset

This result demonstrates that our method also supports high-resolution texture dataset.

Navier–Stokes Fluid Vorticity [B] (2x128x128). We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:

These results demonstrate the applicability of our method on real angular data. Furthermore, our joint amplitude–phase diffusion generates more realistic physical spectra in vector‑valued scientific data, which opens pathways for combined amplitude‑phase generative modeling. We will detail the experimental setup in the supplementary.

Together with the core experiments in the paper, we have now validated this synchronization bias across three distinct domains:

Orientation‑Rich Images (fingerprints, textures mapped as phases), where we demonstrate sharper sample generation with fewer steps,

Earth & Climate Science Data (naturally periodic grids), where we show competitively faithful reconstruction of large‐scale geospatial patterns,

Navier–Stokes Fluid Vorticity (real angular data), where we yield closer matches to the physical energy spectrum when phase coupling is applied.

Across all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where local coherence is critical.

[A] Clifford Neural Layers for PDE Modeling. ICLR 2023.

[B] Hyperspherical Variational Auto-Encoders. UAI 2018.

[C] Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres. ICML 2025.

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We hope these additional results and explanations clarify the versatility and strengths of Kuramoto‑based diffusion, and we are more than happy to answer any further questions!

Very well. Thank you for responding swiftly.

I will gladly increase my score from 3 to 4. Also, I will further my scores on significance and originality.

Could you please include a results for the negative log-likelihood computation then? It would be nice to see how the generated samples align with the training data w.r.t NLL.

Lastly, a final critique, I suggest improving the quality of your images in the main figure. They do look a bit too pixelated. You can also perform additional experiments on ImageNet64 as well, for your final print.

Best of luck to you, the authors!

We thank $\textcolor{brown}{\text{Reviewer X1gf}}$ for their thoughtful critique and recognition of our novelty. Below, we address each question point by point:

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1. Phase Mapping & Validation on True Angular Data

For image datasets, we do linearly map the normalized pixel intensities to phase angles. We agree with $\textcolor{brown}{\text{Reviewer X1gf}}$ that this mapping does not encode spatial relations of pixels. However, it provides a simple, stable, one‑dimensional circular representation: in orientation‑rich images (e.g., fingerprints, textures), coherent ridges and edges manifest as locally consistent intensity patterns, which become locally aligned phases under this mapping.

We considered more sophisticated alternatives but found none that reliably preserved both periodicity and semantic structure:

HSV mapping: We tried mapping RGB images to the HSV color space. However, only the Hue channel is periodic; the other two channels are non-periodic and thus remain untreated.

Spherical Autoencoder: We also tried using a Spherical Autoencoder to encode the pixel intensity into a periodic latent code. However, this approach collapses semantic structure in multi‑class datasets like CIFAR‑10 -- images of different categories fall into similar angles, which greatly hurts sample fidelity.

By contrast, our linear mapping is data-agnostic, easy to implement, and most importantly, lets the Kuramoto drift operate directly on $S^1$, naturally preserving local coherence.

What we really aim to demonstrate in the paper is that the Kuramoto-based diffusion in the circular domain yields a synchronization inductive bias, destroying signals faster and effectively modeling local coherence. Although this phase embedding serves as a good proxy for natural images, this synchronization mechanism is ideally suited for truly angular data. To demonstrate the scalability beyond synthetic phase embeddings, we apply the Kuramoto model to genuinely periodic scientific data:

Navier–Stokes Fluid Vorticity [A] (2x128x128). We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:

These experiments confirm that our Kuramoto diffusion can effectively model real physical angular data, complementing the image‑based experiments. Moreover, the joint amplitude–phase framework opens the door to polar diffusion models on vector‑valued fields. Detailed setup will be included in the supplementary.

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2. Isotropic Baselines & No‑Coupling Variant

To isolate the impact of Kuramoto coupling, we include a model variant with $K(t)=0$, which keeps only the phase-reference drift in the SDE, as well as blurring diffusion models [16,35] with isotropic linear drifts.

FID on Brodatz Textures

Improved Quality in Fewer Steps. Removing Kuramoto coupling raises 100‑step FID by over 15 points, and isotropic baselines [16,35] also incur much higher FIDs. These results confirm that non‑isotropic phase synchronization is the key driver of both rapid convergence and structure‑preserving diffusion on orientation‑rich data.

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3. Choice of $\psi_{ref}$

For the reference phase $\psi_{ref}$, we simply fix it to 0 by default, as it is the middle point of $[-\pi, \pi]$. This value can be tuned to improve alignment with data statistics. For example, for the Brodatz texture images, choosing a slightly negative $\psi_{ref}$ which matches the data distribution can slightly improve the performance.

FID on Brodatz Textures in 100 Steps

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4. 2‑D Embedding vs. Wrapped Gaussians

One can indeed embed the phase via $x,y = \cos(\theta), \sin(\theta) \in \mathbb{R}^2$ and then derive an equivalent SDE in Cartesian coordinates. However, this route introduces several complications. Applying Itô’s lemma with $x,y$ and substituting $d\theta$ yields:

$d x = f_x(x,y)dt + \sqrt{2 D_t} (-y) d W_t, d y = f_y(x,y)dt + \sqrt{2 D_t} (x) d W_t$

where the non-linear drifts $f_x$ and $f_y$ are inherited from Kuramoto coupling in the phase space. The difficulties lie in 3 aspects: (1) The terms in $f_{x}$ and $f_{y}$ involve products like $x_i y_j - y_i x_j$. The mean update is no longer an affine functions of $(x,y)$, (2) the multiplicative noise acts only in the tangent direction, making the covariance dependent on the state $(x,y)$, (3) To ensure $(x,y)$ stay in the unit circle, a projection to $S^1$ has to be performed after every SDE step, which destroys the Gaussian‐increment structure.

Together, these factors mean the exact finite‐step transition in $(x,y)$ is neither Gaussian nor available in closed form. By contrast, we use a wrapped‑Gaussian kernel directly in the angular domain, which offers exact, closed‑form transitions that reside intrinsically on $S^1$, leading to a simple and tractable score‐matching objective.

We will add the full derivations and explanation in the supplementary.

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5. Related Work of Hypersphere Representations & Manifold SDEs

Thank you for highlighting these important lines of research. We will integrate the following into our Related Work section:

Hyperspherical Latent Models: [B] and [C] introduce VAEs with latent spaces on $S^d$, using von Mises–Fisher or power‐spherical priors. These priors generalize the circular von Mises distribution we use on $S^1$.

Diffusion on Rotation & Spherical Manifolds: [D] and [E] extend score‐based diffusion to the 3D rotation group SO(3), deriving trackable marginal kernels and SDE/ODE samplers on these manifolds. [F] presents a linear, homogeneous SDE in a spherical latent space. None of these methods introduces local oscillator‐coupling; their drifts are global and linear.

We differ from previous manifold diffusion methods in the following aspects:

(1) Non-isotropic, Nonlinear, Local Drift. We apply non-isotropic and non-linear Kuramoto coupling that synchronizes local/global phases. In contrast, prior manifold SDEs ([D,E,F]) rely on linear drift fields over the entire manifold.

(2) Intrinsic Wrapped–Gaussian Transitions: We derive exact local transition densities on $S^1$ via wrapped Gaussians and directly learn their scores. Other methods typically embed into $\mathbb{R}^n$ and perform manifold diffusion via manifold projection or auxiliary parametrizations.

(3) Local Coherence Inductive Bias: Crucially, our biologically inspired Kuramoto coupling explicitly aligns neighboring phases, enforcing local coherence (preserving edges, textures) during diffusion. Prior SO(3) and hyperspherical approaches instead prioritize the global symmetry preservation (e.g., rotational invariance) without any mechanism for neighborhood synchronization.

Although both our method and prior diffusion models [D,E,F] leverage circular or hyperspherical geometry, our focus on local phase synchronization is fundamentally different from their emphasis on maintaining geometric invariances. We will cite [B–F] and emphasize these distinctions in the revised manuscript.

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6. Role of the Kuramoto Concept

The Kuramoto concept is the heart of our approach: we borrow the biological phase‐synchronization mechanism -- each oscillator (a phase variable) being pulled toward synchronization with its connected neighbors. By embedding this coupling directly into the forward diffusion SDE, we introduce an inductive bias that actively enforces local alignment, preserving fine‐scale structures (e.g., edges, ridges) as noise is injected.

We validate this synchronization bias across three distinct domains:

Orientation‑Rich Images (fingerprints, textures mapped as phases), where we demonstrate sharper sample generation with fewer steps,

Earth & Climate Science Data (naturally periodic grids), where we show competitively faithful reconstruction of large‐scale geospatial patterns,

Navier–Stokes Fluid Vorticity (real angular data), where we yield closer matches to the physical energy spectrum when phase coupling is applied.

Across all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where local coherence is critical.

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7. Limitations

We thank the reviewer for the reminder. We’ll relocate the Limitations and Broader Impact sections into the main text for the camera‑ready version.

[A] Clifford Neural Layers for PDE Modeling. ICLR 2023.

[B] Hyperspherical Variational Auto-Encoders. UAI 2018.

[C] The Power Spherical distribution. ICML Workshop 2020.

[D] Denoising Diffusion Probabilistic Models on SO(3) for Rotational Alignment. ICLR Workshop 2022.

[E] Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysics. AAAI 2024.

[F] Latent SDEs on Homogeneous Spaces. ICML 2023.

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Thank you for your valuable insights -- we hope this clarifies our contributions and welcome any further questions!

We are grateful to $\textcolor{purple}{\text{Reviewer 3GqZ}}$ for highlighting the innovative aspects of our Kuramoto framework and for careful reading and insightful recommendations. We respond to each of the comments below:

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1. Scaling to High‑Resolution & Vector‑Valued Data

Our initial evaluation on SOCOFing and Brodatz benchmarks was designed to highlight clear gains on canonical orientation‑rich datasets while controlling for compute budget. However, the approach itself is not inherently limited to low resolution. In our ongoing experiments, here we scale the Kuramoto model to higher-resolution textures dataset and real angular data.

High-resolution Ground Terrain [A] (3x128x128). Compared with Broadatz textures, this dataset spans diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules:

FID on Ground Terrain Dataset

Navier–Stokes Fluid Vorticity [B] (2x128x128). We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:

These experiments confirm that our Kuramoto diffusion (i) supports higher‑resolution textures and (ii) effectively models real physical angular data. Furthermore, the combination with amplitude diffusion paves the way for integrated amplitude–phase generative modeling. Details of the experimental setup will be provided in the supplementary material.

Future Work. We plan to extend evaluations to large‑scale medical/biological texture collections (e.g. cellular imaging like JUMPCP [C]) in the future work.

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2. Choice of $\psi_{ref}$, Noise Schedules, and Additional Baselines

Choice of $\psi_{ref}$. For the reference phase $\psi_{ref}$, we simply fix it to 0 by default, as it is the middle point of $[-\pi, \pi]$. $\textcolor{purple}{\text{Reviewer 3GqZ}}$ is right that this value can be tuned to improve alignment with data statistics. For example, for the Brodatz textures images, choosing a slightly negative $\psi_{ref}$ which matches the data distribution can slightly improve the performance.

FID on Brodatz Textures in 100 Steps

Noise Schedules. For the noise and coupling strength, we keep the relation $K_{\text{ref}}(t) > D_t > K(t)$ to ensure that the image information is destroyed in the forward process. In the experiments of our paper, all baselines use a linear noise schedule (with “SGM” = VP‑SDE) -- please see lines 53–61 of the Supplementary for full details. Here we evaluate both VP‑SDE and our model under a cosine schedule. Although switching to cosine reduces FID for all methods -- most notably at 100 steps -- our Kuramoto model still outperforms by a wide margin:

FID on Brodatz Textures

Additional Baselines. We add VE‑SDE and two blurring diffusion models [35, 16], which exhibit similar smoothing behavior. VE‑SDE underperforms VP‑SDE slightly, and the blurring model [16] matches VP‑SDE’s FID. Our Local Kuramoto model maintains a large lead:

FID on Brodatz Textures

These results confirm that, even with different noise schedules and additional baselines, the phase synchronization induced by Kuramoto coupling remains the key driver of improved FID. We will add a succinct discussion of these schedule choices and their quantitative impact in the revised manuscript (Sec. 3.1).

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3. Structured Destruction

By “structured destruction”, we refer to the two-stage structured noising process induced by our Kuraomoto drift: (1) early preservation of coherent features via local phase coupling, and (2) accelerated convergence to the noise distribution at later timesteps. Since non-isotropic Kuramoto coupling enforces local phase alignment, coherent image features (edges, ridges) will stay in synchronization and resist randomization during early noise injection. In later stages, these structures keep synchronized and jointly evolve to noise distribution at a faster pace.

Quantitative Evidence (Fig. 5): The signal‑to‑noise ratio (SNR) can be considered a good proxy metric of how well the main image structure is preserved. Formally, it is defined as:

$SNR(t) = \frac{||x_0 - \mathbb{E}[x_t]||^2}{Var[x_t]}$

which tracks how much of the original structure remains at time $t$. Our Kuramoto diffusion exhibits substantially higher SNR than VP‑SDE over the first 20–30 % of steps, directly quantifying that more “signal” (i.e., structured features) survives before noise overwhelms. After this window, our SNR decays more quickly, reflecting the faster convergence to the noise distribution.

Qualitative Evidence (Fig. 1): The quantitative evidence also correlates with the empirical observation in Fig. 1. Under our Kuramoto model, the samples retain contours and overall shape long after a standard VP‑SDE has already washed them out.

Spectral Interpretation: From a spectral standpoint, local phase coupling behaves like an angular low‑pass filter. In the small‑angle approximation ($\sin(\theta_i-\theta_j)\approx \theta_i-\theta_j$) and without a global reference phase, the forward SDE reduces to a heat‑equation form (or a graph Laplacian equation equivalently):

$\frac{\partial \theta}{\partial t} = K \nabla^2 \theta + \sigma_t\epsilon$

Each Fourier component decays like $e^{-K k^2 t}$, where $k$ denotes the spatial frequency. Modes with very high spatial frequency $k$ (pixel‑scale noise) are damped almost instantly, while moderate‑frequency modes (coherent, edge‑defining structures) decay much more slowly. Consequently, the Kuramoto drift effectively filters away pixel‑scale noise while preserving the smooth, orientation‑rich patterns that constitute edges.

We will add the above discussion of structured destruction in Sec. 3.3 of the revised main paper.

[A] Deep texture manifold for ground terrain recognition. CVPR 2018.

[B] Clifford Neural Layers for PDE Modeling. ICLR 2023.

[C] JUMP Cell Painting dataset: morphological impact of 136,000 chemical and genetic perturbations. bioRxiv 2023.

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Thank you again for your insightful feedback! We hope these revisions will greatly enhance the clarity, and we are more than happy to have further discussions.

We appreciate $\textcolor{red}{\text{Reviewer 4DgT}}$’s thoughtful feedback and the recognition of both the novelty and clarity of our Kuramoto diffusion model. Our responses to the comments follow:

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1. Stronger Justification for “Orientation‑rich” Data

Our non‑isotropic Kuramoto coupling term, $K\sum \sin(\theta_i-\theta_j)$, encourages similar phases to synchronize during the early stages of diffusion. The regions of consistent orientations, which have similar pixel intensities, will remain coherent, resist randomization, and naturally stay in synchronization during early noise injection. This “phase coherence” pulls similar phases together and prevents the rapid decorrelation that destroys edges and structures.

Qualitative Evidence: Fig. 1 of the main paper clearly illustrates the point: samples retain contours and overall shape long after a standard VP‑SDE has already washed them out.

Quantitative Evidence: The signal‑to‑noise ratio (SNR) can be considered a good proxy metric of how well the main image structure is preserved. In Fig. 5 we plot the SNR over forward diffusion timesteps and observe that our SNR stays substantially higher for the first 20–30 % of steps, directly reflecting better preservation of edges and structural details.

Spectral Interpretation: From a spectral standpoint, local phase coupling behaves like an angular low‑pass filter. In the small‑angle approximation ($\sin(\theta_i-\theta_j)\approx \theta_i-\theta_j$) and without a global reference phase, the forward SDE reduces to a heat equation (or a graph Laplacian form equivalently):

$\frac{\partial \theta}{\partial t} = K \nabla^2 \theta + \sigma_t\epsilon$

Each Fourier component decays like $e^{-K k^2 t}$, where $k$ denotes the spatial frequency. Modes with very high spatial frequency $k$ (pixel‑scale noise) are damped almost instantly, while moderate‑frequency modes (coherent, edge‑defining structures) decay much more slowly. Consequently, the Kuramoto drift effectively filters away pixel‑scale noise while preserving the smooth, orientation‑rich patterns that constitute edges.

We will add the above discussion in Sec. 3.3 of the revised main paper.

Note: Although we cannot update the supplementary material at this stage, we will include forward‑only Kuramoto snapshots on fingerprint and texture data in the camera‑ready version to further demonstrate this structure-preserving behavior.

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2. Expanded Baselines & Noise Schedules

In the experiments of our paper, all baselines use a linear noise schedule (with “SGM” referring to VP‑SDE) -- please see lines 53–61 of the supplementary for full details. To isolate the benefit of phase synchronization, we extend our comparisons as follows:

Additional Baselines. We add VE‑SDE and two blurring diffusion models [35, 16], which exhibit similar smoothing behavior. VE‑SDE underperforms VP‑SDE slightly, and the blurring model [16] matches VP‑SDE’s FID. Our Local Kuramoto model maintains a large lead:

FID on Brodatz Textures

Cosine Schedule Ablation. We also evaluate both VP‑SDE and our model under a cosine schedule. Although switching to cosine reduces FID for all methods -- most notably at 100 steps -- our Kuramoto model still outperforms by a wide margin:

FID on Brodatz Textures

These results confirm that, even with different noise schedules and additional baselines, the phase synchronization induced by Kuramoto coupling remains the key driver of improved FID. We will add a succinct discussion of these schedule choices and their quantitative impact in the revised manuscript (Sec. 3.1).

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3. Additional Experiments on High‑Resolution & Vector‑Valued Data

To further validate our method’s generality, we evaluated our method on two new datasets:

High-resolution Ground Terrain [A] (3x128x128). Compared with Brodatz textures, this dataset covers diverse high-resolution materials (e.g., plastic, turf, steel, asphalt, leaves, stone, and brick). We run our Kuramoto diffusion model on this dataset and compare against VP-SDE under the linear schedules:

FID on Ground Terrain Dataset

This result demonstrates that our method also supports high-resolution texture dataset.

Navier–Stokes Fluid Vorticity [B] (2x128x128). We take the PDE dataset from [B] which consists of the fluid vorticity in horizontal and vertical directions at a fixed timestep. We convert the vorticity fields into polar form (amplitude + phase). The amplitude follows standard VP‑SDE, while the phase evolution is driven by our Kuramoto model. We assess generation quality in the spectral domain via the energy‑spectrum mid-frequency slope difference and normalized Wasserstein distance:

These results demonstrate the applicability of our method on real angular data. Furthermore, our joint amplitude–phase diffusion generates more realistic physical spectra in vector‑valued scientific data, which opens pathways for combined amplitude‑phase generative modeling. We will detail the experimental setup in the supplementary.

Together with the core experiments in the paper, we have now validated this synchronization bias across three distinct domains:

Orientation‑Rich Images (fingerprints, textures mapped as phases), where we demonstrate sharper sample generation with fewer steps,

Earth & Climate Science Data (naturally periodic grids), where we show competitively faithful reconstruction of large‐scale geospatial patterns,

Navier–Stokes Fluid Vorticity (real angular data), where we yield closer matches to the physical energy spectrum when phase coupling is applied.

Across all settings, this synchronization bias proves to be a powerful, domain‑agnostic mechanism for modeling data where local coherence is critical.

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4. Richer Qualitative Comparisons

Although we cannot update the current supplementary for now, in the camera‑ready version we will add side‑by‑side, zoomed‐in comparisons of fingerprint and texture samples that clearly show crisper ridge patterns under our Kuramoto diffusion versus the standard diffusion baseline.

[A] Deep texture manifold for ground terrain recognition. CVPR 2018.

[B] Clifford Neural Layers for PDE Modeling. ICLR 2023.

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We appreciate $\textcolor{red}{\text{Reviewer 4DgT}}$’s insightful feedback and hope these clarifications and extended evaluations will convincingly demonstrate the advantages of our Kuramoto diffusion model. Any further comments are more than welcome!