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Colored Noise Diffusion Sampling

Hadar Davidson, Noam Issachar, Sagie Benaim

arXiv:2605.30332v1cs.CV

TL;DR

Diffusion SDE samplers ignore the models’ time-varying spectral bias by injecting uniform white noise. CNS replaces this with dynamic, frequency-aware energy allocation and improves FID across diverse architectures, including SiT-XL/2 from 8.26 to 6.27 on ImageNet-256.

  • Problem

    Existing diffusion SDE samplers ignore evolving frequency content by uniformly injecting white noise, motivating spectral-bias-aware sampler design.

  • Method

    CNS is a training-free stochastic solver that dynamically allocates variance-preserving injected noise across frequency bands according to timestep and structural resolution.

  • Results

    CNS outperforms standard ODE and SDE baselines across SiT, JiT, and FLUX, reducing SiT-XL/2 ImageNet-256 unguided FID from 8.26 to 6.27.

  • Takeaways & Limitations

    CNS is a plug-and-play inference-time sampler substitution that improves generation fidelity across diverse architectures without additional training.

  • Takeaways & Limitations

    CNS requires an SDE framework and high step budgets, making standard ODE solvers preferable for ultra-fast inference.

Abstract

from arXiv · show

Diffusion models achieve state-of-the-art image synthesis, with their generative trajectories fundamentally exhibiting a spectral bias, resolving low-frequency global structures early and high-frequency fine details later. Conventional stochastic differential equation (SDE) solvers fail to account for this dynamic, naively injecting uniform white noise throughout the entire process and misusing the finite energy budget. In this work, we establish a mathematical framework that reconsiders SDE inference as a targeted, frequency-decoupled energy transfer. Leveraging this framework, we introduce Colored Noise Sampling (CNS), a novel, training-free stochastic solver. Rather than injecting uniform white noise, CNS utilizes a dynamic, timestep- and frequency-dependent schedule that more efficiently allocates injected energy toward structurally unresolved frequency bands. By actively exploiting the model's inherent spectral bias, CNS systematically steers the generated distribution toward the true data manifold. Extensive experiments demonstrate that CNS significantly outperforms standard ODE and SDE baselines as a strictly plug-and-play, inference-time sampler substitution across diverse architectures (SiT, JiT, FLUX). Compared to standard sampling on ImageNet-256, CNS achieves substantial unguided FID reductions, improving from 8.26 to 6.27 on SiT-XL/2, 32.39 to 26.69 on JiT-B/16, and 11.88 to 8.31 on JiT-H/16, while yielding consistent relative FID improvements with Classifier-Free Guidance. Project page is available at https://hadardavidson.github.io/CNS/.

1 Introduction

Diffusion sampling exhibits a spectral bias, resolving low-frequency structure before high-frequency detail, while standard SDE methods inject uniform noise without accounting for this evolution. The paper introduces Colored Noise Sampling (CNS), a training-free solver that reallocates variance-preserving stochastic energy toward unresolved frequency bands.

  • Motivation: Diffusion models resolve low-frequency global structures early and high-frequency fine details later, creating a spectrally biased sampling trajectory.This spectral bias is presented as an inherent inductive property of diffusion models.
  • Motivation: Standard SDE samplers inject uniform white noise while disregarding the generated image’s dynamically evolving frequency spectrum.The paper identifies this mismatch as the root inefficiency targeted by its stochastic solver.
  • Framework: The framework treats SDE noise injection as frequency-decoupled energy transfer, with injected noise mapping to spatial features in the same frequency band.Adjustments remain variance-preserving when total injected energy per step stays normalized.
  • Method: CNS is a training-free stochastic solver using a timestep- and frequency-dependent schedule to allocate injected noise energy toward structurally unresolved bands.The schedule is based on how different frequency bands progress during generation.
  • Experiments: Experiments across SiT, JiT, and FLUX show that CNS significantly outperforms standard ODE and SDE baselines, evaluated primarily with FID.The evaluation spans latent-space, pixel-space, and text-to-image generation, under both unguided and guided settings.

2 Related Work

Prior diffusion samplers primarily reduce discretization errors through improved numerical solvers, but remain agnostic to the evolving spectral structure. Spectral-bias methods modify noise distributions, yet many require costly retraining when they alter the learning objective.

  • Samplers for Diffusion Models: Diffusion samplers mitigate discretization errors using higher-order solvers, dynamic solver alternation, and state reparameterizations.These approaches maintain fidelity at low step counts, accelerate generation, or smooth integration pathways.
  • Samplers for Diffusion Models: Existing samplers remain agnostic to the evolving spectral structure of diffusion-model generation.
  • Leveraging Spectral Bias in Diffusion Models: Spectral-bias methods alter initial or temporally evolving noise distributions, including formally grounded frequency-dependent processes such as EqualSNR.These methods are primarily developed by changing noise distributions during training.
  • Leveraging Spectral Bias in Diffusion Models: Altering the learning objective fundamentally requires costly model retraining.

3 Method

The method reframes diffusion inference as frequency-aware energy transfer: CNS dynamically colors injected noise to address spectral bias and steer generated spectra toward the true data manifold. It builds on the corrective role of SDEs, preserved noise energy, and the spectral gap of standard samplers.

  • Colored Noise Sampling: CNS shapes the injected noise spectrum over time and frequency, directing energy toward unresolved bands and steering the generated PSD toward the true data manifold.Its dynamic coloring improves injected-energy efficiency and integrates directly into standard SDE solvers through Algorithm 1.
  • Diffusion sampling dynamics: Inference uses reverse-time differential equations with deterministic or stochastic solvers; SDEs counteract discretization drift through noise injection and score-based restoration.Deterministic sampling can drift off the data manifold because numerical and network errors accumulate, whereas SDEs repeatedly correct the trajectory.
  • Inference-time spectral bias: The model’s spectral bias resolves low-frequency structures early, while high-frequency details emerge gradually and materialize near the end of sampling.The progress index γ(f, t) measures how much of each frequency band’s final structure has been resolved at timestep t.
  • Noise energy preservation: Sampling preserves substantial energy from the initial noise, making inference a signal-transfer process that maps retained noise energy onto structured image features.The geometric interpretation associates trajectories with approximately norm-preserving rotations toward the target data manifold.
  • Spectral gap: Standard deterministic and stochastic samplers leave a spectral gap because neither perfectly recovers the true data PSD, and matching spectra alone does not ensure coherent spatial structure.The restored energy must align with the target manifold’s spatial organization for true distributional matching.

4 Experiments

Section 4 evaluates CNS across class-conditional generation, text-to-image generation, solver settings, CFG, alternative noise training, and schedule ablations. Across these settings, CNS improves sampling quality while remaining a plug-and-play inference-time substitution.

  • Class-Conditional Generation: CNS is evaluated on ImageNet-256 across pixel-space x-prediction and latent-space v-prediction models, using FID, sFID, diversity, and manifold-coverage metrics.The comparisons use official pre-trained weights and standard ODE and SDE baselines.
  • Sampling Steps: Once enough sampling steps enable proper stochastic simulation, CNS consistently outperforms standard sampling across time discretizations.Figure 6 reports FID as a function of sampling steps.
  • Solver Order: CNS outperforms all tested solvers on SiT-XL/2 across weak-convergence orders from first-order Euler-Maruyama to second-order Heun, SRK2, and SRK2S.Weak convergence is emphasized because generative metrics evaluate distributional alignment.
  • Classifier-Free Guidance: Under classifier-free guidance, CNS consistently outperforms standard ODE and SDE samplers across SiT-XL/2, JiT-H/16, and JiT-B/16.The comparison uses Euler with 250 steps for SiT and 50 steps for JiT, reporting each method’s best-performing CFG scale when scales differ.
  • Orthogonality to Alternative Noise Training: CNS remains beneficial when integrated with BNDM’s pre-trained IADB models, demonstrating orthogonality to alternative noise-training modifications.BNDM uses a time-evolving white-to-blue noise distribution, while CNS is applied as the sampler.
  • Ablations: The CNS formulation achieves the highest overall fidelity in ablations, while unnormalized variance scaling significantly degrades quality and validates the finite-energy constraint.The ablation study is conducted on SiT-XL/2 with Euler and 250 steps.

5 Conclusion … A.2 Theoretical Demand for Injected Noise Variance Conservation

The paper frames CNS as a training-free sampler that reallocates stochastic energy toward unresolved frequencies, while theory shows the injected energy budget is finite, timestep-invariant, and cannot be globally rescaled without disrupting target-flow matching. CNS remains limited by its reliance on high-step SDE sampling, motivating deterministic extensions and broader applications.

  • 5 Conclusion: CNS replaces uniform white-noise injection with dynamic frequency-dependent energy routing that exploits spectral bias and targets structurally unresolved frequency bands.The method is presented as a strict, training-free stochastic sampler substitution.
  • 5 Conclusion: CNS requires an SDE framework and high step budgets, making standard ODE solvers preferable for ultra-fast inference.Future work targets deterministic low-step extensions and video generation.
  • A Theoretical Constraints on Stochastic Energy Injection: The stochastic process has a finite injected energy budget that is invariant to solver timestep discretization.This premise underpins the theoretical constraints developed in Appendices A.1 and A.2.
  • A.1 Invariance of Total Injected Energy to Timestep Discretization: The total injected variance converges to a finite constant because bounded diffusion schedules are integrated over a finite time interval.The discrete variance sum corresponds to a Riemann sum for g^2(t).
  • A.1 Invariance of Total Injected Energy to Timestep Discretization: O(1/N) convergence means any timestep count draws from the same finite injected-energy pool, aside from bounded numerical integration error.Here N denotes the number of discretization steps.
  • A.2 Theoretical Demand for Injected Noise Variance Conservation: Even with an exact score oracle, globally misaligned injected noise variance prevents Langevin dynamics from converging to the target data distribution.The analysis uses continuous-time SDE–ODE equivalence to compare the perturbed process with the target flow.
  • A.2 Theoretical Demand for Injected Noise Variance Conservation: β^2 = 1 is necessary and sufficient for the modified stochastic process to preserve the target deterministic flow when the score is generically nonzero.Any global rescaling of injected stochastic noise therefore disrupts transport toward pdata.
  • A.2 Theoretical Demand for Injected Noise Variance Conservation: When the generated density differs from the true path, the rigid score-coupled flow routes probability mass incorrectly, producing bottlenecks or overshoots.The mismatch changes spatial gradients while the transport field remains tailored to the true distribution.

A.3 Model Robustness to Out-of-Distribution Spectral States … B.1.1 Pathwise Energy Dynamics in Continuous-Time Sampling

The paper argues that diffusion models robustly process spectrally atypical intermediate states, supporting Colored Noise Sampling, and explains ODE–SDE spectral differences through score-error-driven pathwise energy divergence. Under an ideal score, stochastic heat and score contraction cancel exactly; imperfect score-state correlation breaks this balance.

  • A.3 Model Robustness to Out-of-Distribution Spectral States: CNS assumes frequency-wise SNR modulation via βf(t) does not drive intermediate states beyond the network’s OOD robustness envelope.This assumption is motivated by diffusion models’ inductive spectral bias.
  • A.3 Model Robustness to Out-of-Distribution Spectral States: Standard inference diverges from training marginals: low-frequency bands resolve structural energy early, while high-frequency bands remain noise-dominated later.The empirical progression γactual(f, t) therefore departs substantially from γtarget(f, t) = 1 −t.
  • A.3 Model Robustness to Out-of-Distribution Spectral States: Despite saturated low frequencies and severe PSD/SNR mismatch, the learned score remains stable and pushes OOD intermediate states toward the true image distribution.This robustness supports routing stochastic energy by frequency through βf(t) while conserving the global variance budget.
  • B Theoretical Analysis of Spectral Dynamics and Colored Noise: The theoretical analysis links ODE–SDE spectral differences to divergent pathwise energy trajectories caused by imperfect score approximation.It motivates analyzing when and how frequency-band energy transfers differ between samplers.
  • B.1 Theoretical Origins of the Generated Distributions Spectral Difference: Although Probability Flow ODE and reverse-time SDE share identical marginals pt(x), their pathwise energy dynamics can differ substantially.The analysis defines state energy and derives its progression for deterministic and stochastic trajectories.
  • B.1.1 Pathwise Energy Dynamics in Continuous-Time Sampling: Under the ideal score sθ = s∗, stochastic heat injected by Itô noise is exactly canceled by restorative radial contraction.This establishes the ideal generative energy balance.
  • B.1.1 Pathwise Energy Dynamics in Continuous-Time Sampling: With an imperfect score sθ = s∗ + ϵ, the heat–contraction cancellation fails and creates an energy gap between generated distributions.The divergence is controlled by the state-error correlation term E[x⊤t ϵ(xt, t)].

B.1.2 State-Error Correlation and the Power Spectral Density Gap … B.2.2 Zero-Sum Spectral Constraints under an Ideal Score

The analysis decomposes state–error correlation and shows that SDEs alternately over- or under-allocate spectral energy depending on whether target-band energy is below or above initial-noise energy. Colored noise reshapes band-wise stochastic energy while conserving the global budget, making spectral adjustments a zero-sum redistribution under an ideal score.

  • B.1.2 State-Error Correlation and the Power Spectral Density Gap: Parseval–Plancherel decomposes global state–error correlation into independent frequency contributions.The framework defines a per-frequency correlation functional Γf(t) and relates it to the band-wise energy gap.
  • B.1.2 State-Error Correlation and the Power Spectral Density Gap: In the attenuation regime, where Rf < Nf, weakened score attenuation makes the SDE over-allocate energy relative to the ODE.This regime typically occurs at high frequencies, where the learned score is too weak to dissipate all continuously injected noise.
  • B.1.2 State-Error Correlation and the Power Spectral Density Gap: In the amplification regime, where Rf > Nf, weakened score amplification makes the SDE under-allocate energy relative to the ODE.This regime typically occurs at low frequencies, where stochastic disruption and insufficient score strength leave the trajectory below the ideal energy level.
  • B.1.2 State-Error Correlation and the Power Spectral Density Gap: At the crossover Rf = Nf, radial approximation error vanishes and ODE and SDE energy trajectories match exactly.The score is purely tangential at this frequency, so the state–score and state–error inner products are zero.
  • B.2.1 Energy Dynamics under an Ideal Score Function: Colored noise scales standard stochastic noise differently across frequency bands using positive, time-dependent weights βf(t).The modification is formulated in the Fourier domain and then mapped back to the spatial domain through a linear operator.
  • B.2.1 Energy Dynamics under an Ideal Score Function: CNS preserves the standard SDE’s global injected-energy budget while changing the band-wise Itô heat from 1/2g2(t) to 1/2g2(t)β2_f(t).Under the ideal-score approximation, β2_f(t) > 1 adds energy to a band, whereas β2_f(t) < 1 dissipates it.
  • B.2.2 Zero-Sum Spectral Constraints under an Ideal Score: If βf > 1 increases energy in one band, global conservation requires deviations from the unit baseline to sum to zero.The resulting spectral adjustment is a zero-sum redistribution across frequency bands under an ideal score.

B.2.3 Breaking the Zero-Sum Constraint via Score Approximation Error

The section argues that score-approximation error becomes weakly correlated with resolved structural states, breaking zero-sum energy redistribution and making injection timing decisive. It then motivates CNS’s deficit-proportional, dynamically normalized allocation of variance toward unresolved frequency bands.

  • Proposed Mechanisms for State-Error Correlation Decay: State–error correlation Γf(t) decays as frequency bands become structurally resolved, despite potentially nonzero score-error magnitude.The authors present this as a plausibility argument involving vanishing radial score components and phase-random residual errors.
  • State-Dependent Efficiency and the Breakdown of Zero-Sum Redistribution: When Γf(t) ≈0, surplus injected variance is treated as ambient noise and removed, so global variance conservation does not imply zero-sum generated-energy conservation.This dismantles the zero-sum redistribution constraint for fully built frequency bands.
  • State-Dependent Efficiency and the Breakdown of Zero-Sum Redistribution: Permanent spectral changes depend on injecting variance while bands remain structurally deficient, not on the total volume of injected energy.Colored-noise efficacy is governed by the temporal overlap between variance scaling βf(t) and structural correlation Γf(t).
  • Derivation of the Proposed CNS Allocation Schedule: CNS routes variance toward lagging frequencies with high Γf(t), using a smooth scaling proportional to each band’s structural deficit rather than concentrating all variance in one band.The schedule is intended to maximize structural retention while avoiding local saturation of the score network.
  • Derivation of the Proposed CNS Allocation Schedule: At initialization, CNS reduces to standard uniform white-noise SDE because every frequency band is entirely unbuilt.As generation proceeds, the allocation dynamically reroutes energy across bands.

C Methodological and Experimental Details … D Additional Results and Extended Evaluations

The appendix turns CNS’s spectral-energy framework into reproducible implementation procedures and extends its evaluation across architectures, solvers, and discretization settings. It defines radial Fourier-band projections, documents empirical sampling relaxations, and introduces broader experimental analyses.

  • C Methodological and Experimental Details: The appendix provides methodological and experimental details for reproducing Colored Noise Sampling and deploying it as an inference algorithm.It translates the theoretical premise of spectral energy control into implementation procedures.
  • C.1 Isolation of Radial Spatial Frequency Bands: CNS isolates isotropic radial frequency bands by applying a channel-wise 2D DFT, shifting frequencies around the DC component, and measuring radial distance.This converts directional frequency coordinates into a one-dimensional spatial-scale representation.
  • C.1 Isolation of Radial Spatial Frequency Bands: The continuous radial-frequency range is partitioned into Nb discrete bands, with each frequency coordinate assigned to a mutually exclusive band mask Bb.The masks support statistical analysis across separately defined spectral regions.
  • C.1 Isolation of Radial Spatial Frequency Bands: The projection operator Pb[·] isolates each band by masking the Fourier spectrum and applying the inverse 2D DFT.The operator is defined using the band mask and element-wise spectral multiplication.
  • C.1 Isolation of Radial Spatial Frequency Bands: Hermitian symmetry ensures that isotropically projected real inputs remain strictly real-valued, enabling direct computation of spatial L2 norms and cosine similarities.The radial masks preserve conjugate symmetry during Fourier masking and inversion.
  • C.2 Hardware Configuration and Empirical Relaxations: CNS relaxes its theoretically optimal β(f, t) schedule with progression scaling and dynamic spectral tilting to improve generative stability.Progression scaling uses c > 1, while spectral tilting smooths allocation transitions across adjacent frequency bands.
  • D Additional Results and Extended Evaluations: The appendix extends the main-text results with benchmarks across architectures and sampling settings, high-order SDE solvers, and robustness tests over discretization steps.These evaluations are organized in App. D.1, App. D.2, and App. D.3.

D.1 Extended Generative Benchmarks · D.2 Numerical Integration Schemes for SDEs

Extended benchmarks show CNS consistently surpasses ODE and standard SDE baselines on unguided JiT-H/16 and improves complex FLUX text-to-image synthesis. For SDE inference, distributional weak order is prioritized because generative evaluation targets the correct distribution rather than individual noise paths.

  • D.1 Extended Generative Benchmarks: CNS strictly dominates ODE and standard SDE baselines across all tracked metrics for unguided JiT-H/16 using 100 sampling steps.The evaluation is conducted on ImageNet-256 without Classifier-Free Guidance.
  • D.1 Extended Generative Benchmarks: CNS enhances complex text-to-image synthesis in the extensive GenEval benchmarking of FLUX.Table 9 reports quantitative compositional accuracy scores on FLUX.1-dev, with higher scores indicating better performance.
  • D.2 Numerical Integration Schemes for SDEs: Discretizing the reverse-time SDE makes the numerical solver determine approximation error, requiring Itô-Taylor expansions rather than the standard Taylor approximations used for ODEs.The reverse-time SDE is written as dx = f(x, t)dt + g(x, t)dw.
  • D.2 Numerical Integration Schemes for SDEs: Strong order p measures pathwise accuracy, with single-trajectory expected error scaling as O(h^p) as step size h approaches zero.It quantifies how accurately a solver tracks a specific noise realization.
  • D.2 Numerical Integration Schemes for SDEs: Weak order q measures distributional accuracy, with errors in expectations of smooth test functions scaling as O(h^q).This definition is distinct from strong order because it concerns expectations rather than individual trajectories.
  • D.2 Numerical Integration Schemes for SDEs: Because generative modeling targets the correct distribution and FID measures distributional distances, weak solver order is the dominant quantity of interest.The experiments evaluate solvers with varying weak orders.
  • D.2 Numerical Integration Schemes for SDEs: Euler-Maruyama provides 1st-order weak and 1/2-order strong convergence with 1 function evaluation per step, while Stochastic Heun provides 2nd-order weak convergence with 2 evaluations.Both methods are included among the evaluated SDE solvers.
  • D.2 Numerical Integration Schemes for SDEs: SRK2 achieves weak order 2 and strong order 1 for additive noise using 2 evaluations per step, while SRK2S retains weak order 2 and strong order 1 for general diagonal multiplicative noise with 3 evaluations.Both schemes are derived by Rößler.

D.3 Robustness to Discretization Steps

CNS is numerically robust across discretization steps: its FID decreases monotonically and consistently beats standard SDE, while reaching ODE peak FID with fewer than half as many steps. However, CNS still requires more minimum discretizations than ODEs, and ODE performance becomes non-monotonic at high step counts.

  • D.3 Robustness to Discretization Steps: CNS’s FID decreases monotonically as sampling steps increase, consistently outperforming the standard SDE baseline.This demonstrates robust performance across varying discretization steps.
  • D.3 Robustness to Discretization Steps: CNS matches the peak FID of the ODE sampler using less than half the steps required by standard SDE.The result indicates accelerated stochastic convergence relative to standard SDE.
  • D.3 Robustness to Discretization Steps: At high step counts, the deterministic ODE sampler fails to maintain monotonic improvement, so results beyond 250 standard steps are omitted.This limitation arises from the pre-trained model’s underlying framework.
  • D.3 Robustness to Discretization Steps: Despite accelerating stochastic convergence, CNS requires a larger minimum number of discretizations than ODEs to properly integrate the underlying differential equations.CNS therefore retains a fundamental limitation shared by SDE solvers.

D.4 Comprehensive Ablation Study … D.4.4 Alternative Noise Formulations (mBm)

The comprehensive ablations test CNS’s energy conservation, spectral allocation, temporal alignment, and alternative noise formulations on unguided SiT-XL/2. Results show that deviations from CNS’s unit-energy, frequency- and time-aware design degrade generation quality, while mBm’s rigid spectral shift performs poorly.

  • D.4 Comprehensive Ablation Study: All ablations use unguided SiT-XL/2 with 250 Euler steps on ImageNet-256, evaluated by FID-10K, sFID, and Inception Score.The quantitative results are reported in Table 10.
  • D.4.1 Global Energy Scaling (Validating the Variance Constraint): Energy scaling away from the unit budget sharply degrades fidelity, with lower energy starving stochastic exploration and higher energy wasting variance.Scaling factors ranged from 0.50 to 2.00; 0.90 is described as starving exploration, while 1.05 already increases energy by 5%.
  • D.4.2 Spectral Perturbation and Partial Corruption: Corrupting the optimal β(f, t) mapping with white noise or randomized unit-energy spectra monotonically worsens performance, even when only 25%, 50%, or 100% of timesteps are altered.The randomized spectra preserve the global energy budget, isolating the effect of band-wise allocation.
  • D.4.2 Spectral Perturbation and Partial Corruption: White →Blue (H : 0.5 →0.1) obtains FID-10K 13.46, sFID 25.16, and IS 122.49.These are the reported multifractional Brownian Motion variant results.
  • D.4.3 Temporal Schedule Permutations: Constant, shuffled, and inverted schedules preserve total frequency-wise energy but fail to match CNS’s FID because they inject energy at incorrect timesteps.The ablations therefore test temporal alignment separately from total energy and frequency-wise allocation.
  • D.4.4 Alternative Noise Formulations (mBm): mBm varies its Hurst parameter H(t), shifting continuously between noise colors such as white noise at H = 0.5 and blue noise at H < 0.5.Its spectral density is governed by the exponent H(t) + 1/2 in the harmonizable representation.
  • D.4.4 Alternative Noise Formulations (mBm): mBm lacks CNS’s granular per-band structural awareness because it imposes a rigid, parameterized shift across the entire spectrum.The passage contrasts this limitation with the γ-matrix’s frequency-specific allocation.

D.5 Generalization to Alternative Noise Training · D.6 Additional Visual Comparisons · CNS (Ours) SDE ODE CNS (Ours) SDE ODE

CNS generalizes to alternative noise-training paradigms through a custom sampler tailored to BNDM’s evolving forward process, while additional comparisons evaluate CNS against ODE and SDE baselines under matched noise realizations. The qualitative results frame CNS as combining ODE structure preservation with SDE detail recovery by routing stochastic energy toward unresolved frequency bands.

  • D.5 Generalization to Alternative Noise Training: CNS was evaluated on official pre-trained IADB models from BNDM to test whether the framework remains applicable beyond conventional noise-training paradigms.The BNDM experiments use a unique, temporally evolving noise distribution.
  • D.5 Generalization to Alternative Noise Training: A custom SDE sampler was derived and implemented specifically for BNDM’s temporally evolving forward process.This tailoring was required because the BNDM experiments rely on a unique noise distribution.
  • D.5 Generalization to Alternative Noise Training: Despite the alternative training objective, the evaluated BNDM models still exhibited a pronounced spectral bias.This observation supports applying the framework across distinct noise-training paradigms.
  • D.6 Additional Visual Comparisons: Additional qualitative experiments compare CNS with deterministic ODE and stochastic SDE baselines across generations from diverse ImageNet classes.These results are presented in Fig. 7 and Fig. 8.
  • D.6 Additional Visual Comparisons: All images within each row use identical noise realizations and class prompts, isolating differences caused by the sampling dynamics.The visual comparisons therefore control both stochastic initialization and conditioning within each row.
  • D.6 Additional Visual Comparisons: Standard white-noise SDEs often produce blurry or structurally degraded textures because they struggle with high-frequency spatial structures.This limitation is contrasted with the behavior of ODE and CNS samplers.
  • D.6 Additional Visual Comparisons: ODEs preserve structure but accumulate errors that can yield over-smoothed, artificial appearances.The passage characterizes this as the deterministic baseline’s principal qualitative limitation.
  • CNS (Ours) SDE ODE CNS (Ours) SDE ODE: CNS bridges the SDE–ODE gap by dynamically routing stochastic energy to unresolved frequency bands, producing sharper fine details.Figures 7 and 8 show visual comparisons among CNS, standard SDE, and ODE samples generated with the same seed.
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