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Spectral Progressive Diffusion for Efficient Image and Video Generation

T0 review · 2 major / 2 minor · reviewed 2026-05-21 · grok-4.3

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Pith's one-line read Diffusion models generate images and videos faster by starting at low resolution and growing it as denoising proceeds from low to high frequencies.

desk verdict The paper turns the known low-to-high frequency order in diffusion into a progressive-resolution schedule with spectral noise expansion, claiming training-free speedups on image and video models. read the letter →

arxiv 2605.18736 v2 pith:KLWO234Z submitted 2026-05-18 cs.CV

Howard Xiao , Brian Chao , Lior Yariv , Gordon Wetzstein This is my paper · ORCID
classification cs.CV
keywords diffusionmodelsimagegenerationvideoefficientinferencefrequencydomainprogressiveresolutionspectralnoise
checked against Cost.FunctionalEquationJ_uniquely_calibrated_via_higher_derivative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that diffusion models implicitly build visual content from low frequencies early in denoising to high frequencies later. This ordering makes high-resolution computation wasteful in the initial noisy stages. Spectral Progressive Diffusion exploits the pattern by expanding resolution along the trajectory using a spectral noise mechanism and a schedule taken from the model's power spectrum. The approach works on existing pretrained models either without any training or with a lightweight fine-tuning step, delivering measurable speedups on both image and video generators while visual quality remains intact.

What carries the argument

Spectral noise expansion mechanism that progressively grows resolution along the denoising trajectory according to a schedule derived from the model's power spectrum.

What would settle it

Running the identical pretrained model at full resolution throughout denoising produces images or videos of equal or higher quality in equal or less wall-clock time than the progressive schedule.

Watch

Extended reading notes

Core claim

Diffusion models generate visual content autoregressively in the frequency domain, with low-frequency components appearing earlier in the denoising process and high-frequency details emerging later. High-resolution computation on noise-dominated frequencies is therefore redundant. Spectral Progressive Diffusion progressively grows resolution along the denoising trajectory of pretrained models by means of a spectral noise expansion mechanism and an optimal resolution schedule derived from the model's power spectrum. This framework supports both training-free acceleration and a fine-tuning recipe that further improves efficiency and quality.

Load-bearing premise

High-resolution computation on noise-dominated frequencies is largely redundant.

Editorial extensions

If this is right

  • Pretrained image and video diffusion models can be accelerated without retraining.
  • A lightweight fine-tuning stage yields further gains in speed and output quality.
  • The same frequency-progression logic applies across both static images and temporal video sequences.
  • Compute savings scale with the length of the denoising trajectory and the chosen resolution schedule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The method could be combined with existing sampler accelerations such as fewer steps or distillation to compound speedups.
  • Similar progressive schedules might transfer to other generative paradigms that exhibit frequency ordering during synthesis.
  • Real-time or edge-device deployment becomes more feasible once early low-resolution stages replace full-resolution passes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper introduces Spectral Progressive Diffusion, a framework for accelerating pretrained diffusion models for image and video generation by progressively growing resolution along the denoising trajectory. It develops a spectral noise expansion mechanism and derives an optimal resolution schedule from the model's power spectrum, supporting both training-free acceleration and a fine-tuning recipe, with claims of significant speedups on SOTA models while preserving visual quality.

Significance. If the central claims hold, the work provides a general, practical approach to reducing the computational cost of high-resolution generation without retraining, which is valuable given the expense of diffusion-based image and video models. The emphasis on leveraging implicit frequency-domain structure in pretrained models and the dual training-free/fine-tuning support are strengths that could influence efficiency-focused extensions in generative modeling.

major comments (2)
  1. [§3] §3 (spectral noise expansion mechanism): The claim that the mechanism enables training-free application of the fixed pretrained denoiser requires that the expanded noise at each resolution transition exactly matches the marginal distribution (variance schedule and cross-frequency correlations) of the original high-resolution forward process at that timestep. The abstract and method description do not provide an explicit verification or derivation showing this preservation, raising a correctness risk for the subsequent denoising steps.
  2. [§4] §4 (optimal resolution schedule derivation): The schedule is stated to be derived from the model's power spectrum, but it is unclear whether the derivation operates under the exact forward-process marginals or relies on empirical averages; if the latter, the schedule may introduce model-specific fitting that undermines the generalizability of the training-free speedup claim.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from explicit quantitative results (e.g., speedup factors, FID or perceptual metrics on specific models like Stable Diffusion or video variants) to ground the 'significant speedups' and 'preserving visual quality' claims.
  2. [Method] Notation for the power spectrum and resolution schedule should be defined with equations early in the method section to improve clarity for readers tracking the frequency-domain arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We have carefully addressed each major comment below with clarifications and planned revisions to improve the rigor and clarity of the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (spectral noise expansion mechanism): The claim that the mechanism enables training-free application of the fixed pretrained denoiser requires that the expanded noise at each resolution transition exactly matches the marginal distribution (variance schedule and cross-frequency correlations) of the original high-resolution forward process at that timestep. The abstract and method description do not provide an explicit verification or derivation showing this preservation, raising a correctness risk for the subsequent denoising steps.

    Authors: We thank the referee for identifying this important aspect of the correctness argument. The current manuscript presents the spectral noise expansion mechanism but does not include a self-contained derivation of marginal preservation. In the revised manuscript we will add a formal derivation in §3 showing that the expansion, by construction via the Fourier basis and power-spectrum scaling, exactly reproduces the variance schedule and cross-frequency covariances of the high-resolution forward process at the transition timestep. We will also include a short verification experiment in the appendix that empirically confirms the distributional match before and after expansion. revision: yes

  2. Referee: [§4] §4 (optimal resolution schedule derivation): The schedule is stated to be derived from the model's power spectrum, but it is unclear whether the derivation operates under the exact forward-process marginals or relies on empirical averages; if the latter, the schedule may introduce model-specific fitting that undermines the generalizability of the training-free speedup claim.

    Authors: We appreciate the referee’s concern about the theoretical grounding and generalizability. The derivation in §4 starts from the exact forward-process marginals and uses the power spectrum to identify the timestep at which high-frequency energy falls below a noise-dominated threshold; the schedule is therefore analytic with respect to those marginals. In practice the power spectrum is estimated once from the pretrained model, but this estimation is not a learned fitting procedure and does not alter the underlying marginals. We will expand §4 to make this distinction explicit, add a short proof sketch linking the schedule directly to the marginal variance expressions, and include a brief discussion of why the same procedure applies to any diffusion model whose frequency-generation ordering is consistent with the observed power-spectrum decay. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation chain

full rationale

The paper states it develops a spectral noise expansion mechanism and derives an optimal resolution schedule from the model's power spectrum. This uses the pretrained model's characteristics as an external input for the schedule rather than reducing the central result to a self-referential fit or self-citation by construction. No equations or steps in the abstract demonstrate that a 'prediction' equals its own fitted input or that a uniqueness claim collapses to prior author work. The framework is positioned as training-free acceleration on existing models, with the power spectrum providing independent frequency-domain structure. This is the common case of a self-contained method against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The approach rests on the domain observation that diffusion models generate low frequencies first. The spectral noise expansion is a newly introduced mechanism without independent evidence outside the paper. No explicit free parameters are named in the abstract.

assumptions (1)
  • domain assumption Diffusion models implicitly generate visual content autoregressively in the frequency domain, with low-frequency components generated earlier.
    Directly stated in the abstract as the foundational observation enabling the method.
invented entities (1)
  • spectral noise expansion mechanism
    purpose: To enable progressive resolution growth along the denoising trajectory
    Newly proposed component of the framework; no external falsifiable evidence provided in abstract.

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Cite this review

Pith. "Pith review of Spectral Progressive Diffusion for Efficient Image and Video Generation." pith.science (2026). https://pith.science/paper/KLWO234Z

@misc{pith2026260518736,
  author       = {Pith},
  title        = {Pith review of: Spectral Progressive Diffusion for Efficient Image and Video Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLWO234Z}},
  note         = {Machine review of arXiv:2605.18736}
}
read the original abstract

Diffusion models have been shown to implicitly generate visual content autoregressively in the frequency domain, where low-frequency components are generated earlier in the denoising process while high-frequency details emerge only in later timesteps. This structure offers a natural opportunity for efficient generation, as high-resolution computation on noise-dominated frequencies is largely redundant. We propose Spectral Progressive Diffusion, a general framework that progressively grows resolution along the denoising trajectory of pretrained diffusion models. To this end, we develop a spectral noise expansion mechanism and derive an optimal resolution schedule from the model's power spectrum. Our framework supports training-free acceleration and a novel fine-tuning recipe that further improves efficiency and quality. We demonstrate significant speedups on state-of-the-art pretrained image and video generation models while preserving visual quality.

Figures

Figures reproduced from arXiv: 2605.18736 by the authors.

Figure 1
Figure 1. Spectral Progressive Diffusion. We progressively grow the resolution along the denoising trajectory using an optimal resolution schedule derived from the spectral power of pretrained models (left). At each scheduled transition, our spectral noise expansion mechanism (right) injects high￾frequency noise at the correct level while preserving the partially-denoised low-frequency content. denoised low-frequency content.… view at source ↗
Figure 2
Figure 2. Diffusion process in the spectral domain. Latent power spectra in both image and video models decay rapidly with frequency (Fig. (a)), consistent with natural images. Diffusion exhibits a frequency-domain autoregressive structure (Fig. (b)) due to the aforementioned property: low frequencies emerge early in the denoising process, while high frequencies remain noise-dominated. 4 Spectral Progressive Diffusion In this… view at source ↗
Figure 3
Figure 3. Visual Generation Qualitative Comparisons. For the main comparison of latent-space image generation, our method outperforms the state-of-the-art spatial acceleration method RALU [32] in both visual fidelity and inference speed. Across all evaluated modalities (latent/pixel-space image generation and latent-space video generation), we achieve substantial acceleration over standard high-resolution baselines while pres… view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: (a): Ablation Studies on δ, S and TΦ. We observe a clear tradeoff between image quality and efficiency when varying δ and S as shown in the top plot. Across transforms, DCT achieves similar quality as DWT and outperforms FFT as shown in the bottom plot. (b): Frequency-…
Figure 5
Figure 5. Figure 5: Spectral noise passthrough experiment. At smaller δ ∈ [0.0001, 0.001], there is almost no observable difference compared to native full-resolution generation. As larger δ values cause high-frequency replacement to persist later in the denoising trajectory, we observe i…
Figure 6
Figure 6. Figure 6: Qualitative comparisons on latent-space image generation. We compare our method against default-step generation, reduced-step native-resolution generation on FLUX.1-dev [39], and RALU [32], a state-of-the-art acceleration baseline matched to similar speedups. Our metho…
Figure 7
Figure 7. Figure 7: Qualitative comparisons on latent-space image generation. We compare our method against default-step generation, reduced-step native-resolution generation on FLUX.1-dev [39], and RALU [32], a state-of-the-art acceleration baseline matched to similar speedups. Our metho…
Figure 8
Figure 8. Figure 8: Qualitative comparisons on latent-space image generation. We compare our method against default-step generation, reduced-step native-resolution generation on FLUX.1-dev [39], and RALU [32], a state-of-the-art acceleration baseline matched to similar speedups. Our metho…
Figure 9
Figure 9. Figure 9: Qualitative comparisons on latent-space image generation (fine-tuned). We compare our method against default-step generation, reduced-step native-resolution generation on Z-Image [5] matched to similar speedups. Our fine-tuned model (Ours∗ ) achieves even higher image …
Figure 10
Figure 10. Figure 10: Qualitative comparisons on latent-space image generation (fine-tuned). We compare our method against default-step generation, reduced-step native-resolution generation on Z-Image [5] matched to similar speedups. Our fine-tuned model (Ours∗ ) achieves even higher image…
Figure 11
Figure 11. Figure 11: Qualitative comparisons on latent-space image generation (fine-tuned). We compare our method against default-step generation, reduced-step native-resolution generation on Z-Image [5] matched to similar speedups. Our fine-tuned model (Ours∗ ) achieves even higher image…
Figure 12
Figure 12. Figure 12: Qualitative comparisons on pixel-space image generation. We compare our method against default-step generation and reduced-step native-resolution generation on PixelGen [55], matched to comparable speedups. An asterisk (Ours∗ ) marks the fine-tuned model. Our method a…
Figure 13
Figure 13. Figure 13: Qualitative comparisons on pixel-space image generation. We compare our method against default-step generation and reduced-step native-resolution generation on PixelGen [55], matched to comparable speedups. An asterisk (Ours∗ ) marks the fine-tuned model. Our method a…
Figure 14
Figure 14. Figure 14: Qualitative comparisons on pixel-space image generation. We compare our method against default-step generation and reduced-step native-resolution generation on PixelGen [55], matched to comparable speedups. An asterisk (Ours∗ ) marks the fine-tuned model. Our method a…
Figure 15
Figure 15. Figure 15: Qualitative ablation on TΦ. We see that FFT leads to overly smooth results while DCT and DWT attain similar image quality. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_15.png]
Figure 16
Figure 16. Figure 16: Qualitative ablation on δ. We observe that increasing δ improves efficiency, but results in ghosting and halo artifacts near detailed edges. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_16.png]
Figure 17
Figure 17. Figure 17: Qualitative ablation on S. We find that increasing S leads to marginal speedup improve￾ments and little image quality degradation. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_17.png]
Figure 18
Figure 18. Figure 18: Texture editing results. Our frequency-based editing framework outperforms SDEdit, enabling high-fidelity texture transfer while preserving the geometric structure of the input image. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_18.png]
Figure 19
Figure 19. Figure 19: Texture editing results. Our frequency-based editing framework outperforms SDEdit, enabling high-fidelity texture transfer while preserving the geometric structure of the input image. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_19.png]
Figure 20
Figure 20. Figure 20: Texture editing results. Our frequency-based editing framework outperforms SDEdit, enabling high-fidelity texture transfer while preserving the geometric structure of the input image. 47 [PITH_FULL_IMAGE:figures/full_fig_p047_20.png]
Figure 21
Figure 21. Figure 21: Texture editing results. Our frequency-based editing framework outperforms SDEdit, enabling high-fidelity texture transfer while preserving the geometric structure of the input image. 48 [PITH_FULL_IMAGE:figures/full_fig_p048_21.png]
Figure 22
Figure 22. Figure 22: Artistic stylization results. Aside from texture editing, our frequency-based editing approach also supports artistic stylization given stylistic descriptions and a representative artist. 49 [PITH_FULL_IMAGE:figures/full_fig_p049_22.png]
Figure 23
Figure 23. Figure 23: Effect of TΦ on image editing. FFT–based editing leads to overly-smooth and hazy results; DCT- and DWT-based editing achieve similar editing quality. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_23.png]

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Reference graph

Works this paper leans on

114 extracted references · 114 canonical work pages · cited by 3 Pith papers

  1. [1]

    Ahmed, T

    N. Ahmed, T. Natarajan, and K. R. Rao. Discrete cosine transform.IEEE Transactions on Computers, C-23(1):90–93, 1974

    work page 1974

  2. [2]

    Longformer: The Long-Document Transformer

    Iz Beltagy, Matthew E Peters, and Arman Cohan. Longformer: The long-document transformer, 2020. URLhttps://arxiv.org/abs/2004.05150

    work page Pith review arXiv 2020

  3. [3]

    Spectral analysis of diffusion models with application to schedule design

    Roi Benita, Miki Elad, and Joseph Keshet. Spectral analysis of diffusion models with application to schedule design. InAdv. Neural Inform. Process. Syst., volume 38, pages 2073–2127, 2026

    work page

  4. [4]

    Token merging for fast stable diffusion

    Daniel Bolya and Judy Hoffman. Token merging for fast stable diffusion. InIEEE Conf. Comput. Vis. Pattern Recog., pages 4599–4603, 2023

    work page 2023

  5. [5]

    Z-Image: An efficient image generation foundation model with single-stream diffusion transformer, 2025

    Huanqia Cai, Sihan Cao, Ruoyi Du, Peng Gao, Steven Hoi, et al. Z-Image: An efficient image generation foundation model with single-stream diffusion transformer, 2025. URL https://arxiv.org/abs/2511. 22699

    work page 2025

  6. [6]

    Chandran, N

    Satish Chandran, Nicolas Roque dos Santos, Yunshu Wu, Greg Ver Steeg, and Evangelos Papalexakis. Spectral regularization for diffusion models, 2026. URLhttps://arxiv.org/abs/2603.02447

    work page arXiv 2026

  7. [7]

    arXiv preprint arXiv:2406.01125 (2024)

    Pengtao Chen, Mingzhu Shen, Peng Ye, Jianjian Cao, Chongjun Tu, Christos-Savvas Bouganis, Yiren Zhao, and Tao Chen. δ-DiT: A training-free acceleration method tailored for diffusion transformers, 2024. URLhttps://arxiv.org/abs/2406.01125

    work page arXiv 2024

  8. [8]

    Rethinking attention with performers

    Krzysztof Choromanski, Valerii Likhosherstov, David Dohan, Xingyou Song, Andreea Gane, Tamas Sarlos, Peter Hawkins, Jared Davis, Afroz Mohiuddin, Lukasz Kaiser, et al. Rethinking attention with performers. InInt. Conf. Learn. Represent., 2021

    work page 2021

Show all 114 references
  1. [9]

    Diffusion is spectral autoregression

    Sander Dieleman. Diffusion is spectral autoregression. Blog post, September 2024. URL https: //sander.ai/2024/09/02/spectral-autoregression.html

    2024

  2. [10]

    DemoFusion: Democratising high-resolution image generation with no $$$

    Ruoyi Du, Dongliang Chang, et al. DemoFusion: Democratising high-resolution image generation with no $$$. InIEEE Conf. Comput. Vis. Pattern Recog., 2024

    2024

  3. [11]

    Flow along the k-amplitude for generative modeling, 2025

    Weitao Du, Shuning Chang, Jiasheng Tang, Yu Rong, Fan Wang, and Shengchao Liu. Flow along the k-amplitude for generative modeling, 2025. URLhttps://arxiv.org/abs/2504.19353

    2025

  4. [12]

    Scaling rectified flow transformers for high-resolution image synthesis

    Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, Dustin Podell, Tim Dockhorn, Zion English, and Robin Rombach. Scaling rectified flow transformers for high-resolution image synthes...

    2024

  5. [13]

    Spectrally-guided diffusion noise schedules

    Carlos Esteves and Ameesh Makadia. Spectrally-guided diffusion noise schedules. InInt. Conf. Mach. Learn., 2026

    2026

  6. [14]

    A fourier space perspective on diffusion models, 2025

    Fabian Falck, Teodora Pandeva, Kiarash Zahirnia, Rachel Lawrence, Richard Turner, Edward Meeds, Javier Zazo, and Sushrut Karmalkar. A fourier space perspective on diffusion models, 2025. URL https://arxiv.org/abs/2505.11278

    2025

  7. [15]

    Vchitect-2.0: Parallel transformer for scaling up video diffusion models,

    Weichen Fan, Chenyang Si, Junhao Song, Zhenyu Yang, Yinan He, Long Zhuo, Ziqi Huang, Ziyue Dong, Jingwen He, Dongwei Pan, et al. Vchitect-2.0: Parallel transformer for scaling up video diffusion models,

  8. [16]

    URLhttps://arxiv.org/abs/2501.08453

  9. [17]

    Attend to not attended: Structure-then-detail token merging for post-training dit acceleration

    Haipeng Fang, Sheng Tang, Juan Cao, Enshuo Zhang, Fan Tang, and Tong-Yee Lee. Attend to not attended: Structure-then-detail token merging for post-training dit acceleration. InIEEE Conf. Comput. Vis. Pattern Recog., pages 18083–18092, 2025

    2025

  10. [18]

    Firmin Didot, 1822

    Jean-Baptiste Joseph Fourier.Théorie Analytique de la Chaleur. Firmin Didot, 1822

  11. [19]

    GenEval: An object-focused framework for evaluating text-to-image alignment

    Dhruba Ghosh, Hannaneh Hajishirzi, and Ludwig Schmidt. GenEval: An object-focused framework for evaluating text-to-image alignment. InAdv. Neural Inform. Process. Syst., 2023

    2023

  12. [20]

    Matryoshka diffusion models

    Jiatao Gu, Shuangfei Zhai, Yizhe Zhang, Joshua M Susskind, and Navdeep Jaitly. Matryoshka diffusion models. InInt. Conf. Learn. Represent., 2023

    2023

  13. [21]

    Make a cheap scaling: A self-cascade diffusion model for higher-resolution adaptation

    Lanqing Guo, Yingqing He, Haoxin Chen, Menghan Xia, Xiaodong Cun, Yufei Wang, Siyu Huang, Yong Zhang, Xintao Wang, Qifeng Chen, Ying Shan, and Bihan Wen. Make a cheap scaling: A self-cascade diffusion model for higher-resolution adaptation. InEur. Conf. Comput. Vis., 2024. 11

    2024

  14. [22]

    Wavelet score-based generative modeling

    Florentin Guth, Simon Coste, Valentin De Bortoli, and Stephane Mallat. Wavelet score-based generative modeling. InAdv. Neural Inform. Process. Syst., volume 35, pages 478–491, 2022

    2022

  15. [23]

    Zur theorie der orthogonalen funktionensysteme.Mathematische Annalen, 69(3):331–371, 1910

    Alfréd Haar. Zur theorie der orthogonalen funktionensysteme.Mathematische Annalen, 69(3):331–371, 1910

    1910

  16. [24]

    Infinity: Scaling bitwise AutoRegressive modeling for high-resolution image synthesis

    Jian Han, Jinlai Liu, Yi Jiang, Bin Yan, Yuqi Zhang, Zehuan Yuan, Bingyue Peng, and Xiaobing Liu. Infinity: Scaling bitwise AutoRegressive modeling for high-resolution image synthesis. InIEEE Conf. Comput. Vis. Pattern Recog., 2025

    2025

  17. [25]

    Agglomerative token clustering

    Joakim Bruslund Haurum, Sergio Escalera, Graham W Taylor, and Thomas B Moeslund. Agglomerative token clustering. InEur. Conf. Comput. Vis., pages 200–218, 2024

    2024

  18. [26]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. InAdv. Neural Inform. Process. Syst., 2020

    2020

  19. [27]

    Cascaded diffusion models for high fidelity image generation.J

    Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. Cascaded diffusion models for high fidelity image generation.J. Mach. Learn. Res., 23(47):1–33, 2022

    2022

  20. [28]

    Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen

    Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. LoRA: Low-rank adaptation of large language models. InInt. Conf. Learn. Represent., 2022

    2022

  21. [29]

    T2I-CompBench: A comprehensive benchmark for open-world compositional text-to-image generation

    Kaiyi Huang, Kaiyue Sun, Enze Xie, Zhenguo Li, and Xihui Liu. T2I-CompBench: A comprehensive benchmark for open-world compositional text-to-image generation. InAdv. Neural Inform. Process. Syst., 2023

    2023

  22. [30]

    Wavedm: Wavelet-based diffusion models for image restoration.IEEE Trans

    Yi Huang, Jiancheng Huang, Jianzhuang Liu, Mingfu Yan, Yu Dong, Jiaxi Lv, Chaoqi Chen, and Shifeng Chen. Wavedm: Wavelet-based diffusion models for image restoration.IEEE Trans. Multimedia, 26: 7058–7073, 2024

    2024

  23. [31]

    Spectralar: Spectral autoregressive visual generation

    Yuanhui Huang, Weiliang Chen, Wenzhao Zheng, Yueqi Duan, Jie Zhou, and Jiwen Lu. Spectralar: Spectral autoregressive visual generation. InInt. Conf. Comput. Vis., pages 15842–15852, 2025

    2025

  24. [32]

    VBench: Comprehensive benchmark suite for video generative models

    Ziqi Huang, Yinan He, Jiashuo Yu, Fan Zhang, Chenyang Si, Yuming Jiang, Yuanhan Zhang, Tianxing Wu, Qingyang Jin, Nattapol Chanpaisit, Yaohui Wang, Xinyuan Chen, Limin Wang, Dahua Lin, Yu Qiao, and Ziwei Liu. VBench: Comprehensive benchmark suite for video generative models. I...

    2024

  25. [33]

    Training-free mixed-resolution latent upsampling for spatially accelerated diffusion transformers, 2026

    Wongi Jeong, Kyungryeol Lee, Hoigi Seo, and Se Young Chun. Training-free mixed-resolution latent upsampling for spatially accelerated diffusion transformers, 2026. URLhttps://arxiv.org/abs/2507. 08422

    2026

  26. [34]

    Pyramidal flow matching for efficient video generative modeling

    Yang Jin, Zhicheng Sun, Ningyuan Li, Kun Xu, Hao Jiang, Nan Zhuang, Quzhe Huang, Yang Song, Yadong Mu, and Zhouchen Lin. Pyramidal flow matching for efficient video generative modeling. InInt. Conf. Learn. Represent., 2025

    2025

  27. [35]

    Subspace diffusion generative models

    Bowen Jing, Gabriele Corso, Renato Berlinghieri, and Tommi Jaakkola. Subspace diffusion generative models. InEur. Conf. Comput. Vis., pages 274–289, 2022

    2022

  28. [36]

    Kapfer, K

    C. Kapfer, K. Stine, B. Narasimhan, C. Mentzel, and E. Candès. Marlowe: Stanford’s GPU-based computational instrument, 2025. URLhttps://doi.org/10.5281/zenodo.14751899

    2025 doi

  29. [37]

    Transformers are rnns: Fast autoregressive transformers with linear attention

    Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pappas, and François Fleuret. Transformers are rnns: Fast autoregressive transformers with linear attention. InInt. Conf. Mach. Learn., pages 5156–5165. PMLR, 2020

    2020

  30. [38]

    Token fusion: Bridging the gap between token pruning and token merging

    Minchul Kim, Shangqian Gao, Yen-Chang Hsu, Yilin Shen, and Hongxia Jin. Token fusion: Bridging the gap between token pruning and token merging. InProceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1383–1392, 2024

    2024

  31. [39]

    DiffuseHigh: Training-free progressive high-resolution image synthesis through structure guidance

    Younghyun Kim, Geunmin Hwang, Junyu Zhang, and Eunbyung Park. DiffuseHigh: Training-free progressive high-resolution image synthesis through structure guidance. InAAAI, 2025

    2025

  32. [40]

    Kingma, Tim Salimans, Ben Poole, and Jonathan Ho

    Diederik P. Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. InAdv. Neural Inform. Process. Syst., 2021

    2021

  33. [41]

    Black Forest Labs. FLUX. Software repository, 2024. URL https://github.com/ black-forest-labs/flux. 12

    2024

  34. [42]

    FLUX.2: Frontier Visual Intelligence

    Black Forest Labs. FLUX.2: Frontier Visual Intelligence. Blog post, 2025. URL https://bfl.ai/ blog/flux-2

    2025

  35. [43]

    Beta sampling is all you need: Efficient image generation strategy for diffusion models using stepwise spectral analysis, 2024

    Haeil Lee, Hansang Lee, Seoyeon Gye, and Junmo Kim. Beta sampling is all you need: Efficient image generation strategy for diffusion models using stepwise spectral analysis, 2024. URL https: //arxiv.org/abs/2407.12173

    2024 arXiv

  36. [44]

    Local representative token guided merging for text-to-image generation, 2025

    Min-Jeong Lee, Hee-Dong Kim, and Seong-Whan Lee. Local representative token guided merging for text-to-image generation, 2025. URLhttps://arxiv.org/abs/2507.12771

    2025

  37. [45]

    Frecas: Efficient higher-resolution image generation via frequency-aware cascaded sampling

    Ruihuang Li, Lei Zhang, et al. Frecas: Efficient higher-resolution image generation via frequency-aware cascaded sampling. InInt. Conf. Learn. Represent., volume 2025, pages 6400–6412, 2025

    2025

  38. [46]

    Radial attention: O(nlogn) sparse attention with energy decay for long video generation

    Xingyang Li, Muyang Li, Tianle Cai, Haocheng Xi, Shuo Yang, Yujun Lin, Lvmin Zhang, Songlin Yang, Jinbo Hu, Kelly Peng, et al. Radial attention: O(nlogn) sparse attention with energy decay for long video generation. InAdv. Neural Inform. Process. Syst., 2025

    2025

  39. [47]

    Wf-vae: Enhancing video vae by wavelet-driven energy flow for latent video diffusion model

    Zongjian Li, Bin Lin, Yang Ye, Liuhan Chen, Xinhua Cheng, Shenghai Yuan, and Li Yuan. Wf-vae: Enhancing video vae by wavelet-driven energy flow for latent video diffusion model. InIEEE Conf. Comput. Vis. Pattern Recog., pages 17778–17788, 2025

    2025

  40. [48]

    Lawrence Zitnick

    Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C. Lawrence Zitnick. Microsoft COCO: Common objects in context. InEur. Conf. Comput. Vis., 2014

    2014

  41. [49]

    Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. InInt. Conf. Learn. Represent., 2023

    2023

  42. [50]

    Timestep embedding tells: It’s time to cache for video diffusion model, 2024

    Feng Liu, Shiwei Zhang, Xiaofeng Wang, Yujie Wei, Haonan Qiu, Yuzhong Zhao, Yingya Zhang, Qixiang Ye, and Fang Wan. Timestep embedding tells: It’s time to cache for video diffusion model, 2024. URL https://arxiv.org/abs/2411.19108

    2024

  43. [51]

    FreqCa: Accelerating diffusion models via frequency-aware caching, 2025

    Jiacheng Liu, Peiliang Cai, Qinming Zhou, Yuqi Lin, Deyang Kong, Benhao Huang, Yupei Pan, Haowen Xu, Chang Zou, Junshu Tang, Shikang Zheng, and Linfeng Zhang. FreqCa: Accelerating diffusion models via frequency-aware caching, 2025. URLhttps://arxiv.org/abs/2510.08669

    2025

  44. [52]

    From reusing to forecasting: Accelerating diffusion models with TaylorSeers

    Jiacheng Liu, Chang Zou, Yuanhuiyi Lyu, Junjie Chen, and Linfeng Zhang. From reusing to forecasting: Accelerating diffusion models with TaylorSeers. InInt. Conf. Comput. Vis., 2025

    2025

  45. [53]

    Flow straight and fast: Learning to generate and transfer data with rectified flow

    Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. InInt. Conf. Learn. Represent., 2023

    2023

  46. [54]

    ToMA: Token merge with attention for diffusion models

    Wenbo Lu, Shaoyi Zheng, Yuxuan Xia, and Shengjie Wang. ToMA: Token merge with attention for diffusion models. InInt. Conf. Mach. Learn., 2025

    2025

  47. [55]

    Latent consistency models: Synthesizing high-resolution images with few-step inference, 2023

    Simian Luo, Yiqin Tan, Longbo Huang, Jian Li, and Hang Zhao. Latent consistency models: Synthesizing high-resolution images with few-step inference, 2023. URLhttps://arxiv.org/abs/2310.04378

    2023 arXiv

  48. [56]

    Deepcache: Accelerating diffusion models for free

    Xinyin Ma, Gongfan Fang, and Xinchao Wang. Deepcache: Accelerating diffusion models for free. In IEEE Conf. Comput. Vis. Pattern Recog., 2024

    2024

  49. [57]

    PixelGen: Pixel diffusion beats latent diffusion with perceptual loss, 2026

    Zehong Ma, Ruihan Xu, and Shiliang Zhang. PixelGen: Pixel diffusion beats latent diffusion with perceptual loss, 2026. URLhttps://arxiv.org/abs/2602.02493

    2026 arXiv

  50. [58]

    SDEdit: Guided image synthesis and editing with stochastic differential equations

    Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. SDEdit: Guided image synthesis and editing with stochastic differential equations. InInt. Conf. Learn. Represent., 2022

    2022

  51. [59]

    completely blind

    Anish Mittal, Rajiv Soundararajan, and Alan C. Bovik. Making a “completely blind” image quality analyzer.IEEE Sign. Process. Letters, 20(3):209–212, 2013. doi: 10.1109/LSP.2012.2227726

    2013 doi

  52. [60]

    Scale space diffusion, 2026

    Soumik Mukhopadhyay, Prateksha Udhayanan, and Abhinav Shrivastava. Scale space diffusion, 2026. URLhttps://arxiv.org/abs/2603.08709

    2026

  53. [61]

    DCTdiff: Intriguing properties of image generative modeling in the DCT space

    Mang Ning, Mingxiao Li, Jianlin Su, Jia Haozhe, Lanmiao Liu, Martin Benes, Wenshuo Chen, Albert Ali Salah, and Itir Onal Ertugrul. DCTdiff: Intriguing properties of image generative modeling in the DCT space. InInt. Conf. Mach. Learn., volume 267, pages 46498–46524. PMLR, 2025. 13

    2025

  54. [62]

    NVIDIA, Yuval Atzmon, Maciej Bala, Yogesh Balaji, Tiffany Cai, Yin Cui, Jiaojiao Fan, Yunhao Ge, Siddharth Gururani, Jacob Huffman, Ronald Isaac, Pooya Jannaty, Tero Karras, Grace Lam, J. P. Lewis, Aaron Licata, Yen-Chen Lin, Ming-Yu Liu, Qianli Ma, Arun Mallya, Ashlee Martino...

    2024

  55. [63]

    Certain topics in telegraph transmission theory.Transactions of the American Institute of Electrical Engineers, 47(2):617–644, 1928

    Harry Nyquist. Certain topics in telegraph transmission theory.Transactions of the American Institute of Electrical Engineers, 47(2):617–644, 1928

    1928

  56. [64]

    Scalable diffusion models with transformers

    William Peebles and Saining Xie. Scalable diffusion models with transformers. InInt. Conf. Comput. Vis., 2023

    2023

  57. [65]

    Wavelet diffusion models are fast and scalable image generators

    Hao Phung, Quan Dao, and Anh Tran. Wavelet diffusion models are fast and scalable image generators. In IEEE Conf. Comput. Vis. Pattern Recog., 2023

    2023

  58. [66]

    DiMSUM: Diffusion Mamba – a scalable and unified spatial-frequency method for image generation

    Hao Phung, Quan Dao, Trung Dao, Hoang Phan, Dimitris Metaxas, and Anh Tran. DiMSUM: Diffusion Mamba – a scalable and unified spatial-frequency method for image generation. InAdv. Neural Inform. Process. Syst., 2024

    2024

  59. [67]

    FreeScale: Unleashing the resolution of diffusion models via tuning-free scale fusion

    Haonan Qiu et al. FreeScale: Unleashing the resolution of diffusion models via tuning-free scale fusion. In Int. Conf. Comput. Vis., 2025

    2025

  60. [68]

    FlowAR: Scale-wise autoregressive image generation meets flow matching

    Sucheng Ren, Qihang Yu, Ju He, Xiaohui Shen, Alan Yuille, and Liang-Chieh Chen. FlowAR: Scale-wise autoregressive image generation meets flow matching. InInt. Conf. Mach. Learn., 2025

    2025

  61. [69]

    Generative modelling with inverse heat dissipation

    Severi Rissanen, Markus Heinonen, and Arno Solin. Generative modelling with inverse heat dissipation. InInt. Conf. Learn. Represent., 2023

    2023

  62. [70]

    Origins of scaling in natural images.Vis

    Daniel L Ruderman. Origins of scaling in natural images.Vis. Res., 37(23):3385–3398, 1997

    1997

  63. [71]

    Pyramidal denoising diffusion probabilistic models, 2022

    Dohoon Ryu and Jong Chul Ye. Pyramidal denoising diffusion probabilistic models, 2022. URL https: //arxiv.org/abs/2208.01864

    2022

  64. [72]

    Progressive distillation for fast sampling of diffusion models

    Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. InInt. Conf. Learn. Represent., 2022

    2022

  65. [73]

    Claude E. Shannon. Communication in the presence of noise.Proceedings of the IRE, 37(1):10–21, 1949

    1949

  66. [74]

    Latent wavelet diffusion for ultra-high-resolution image synthesis, 2025

    Luigi Sigillo, Shengfeng He, and Danilo Comminiello. Latent wavelet diffusion for ultra-high-resolution image synthesis, 2025. URLhttps://arxiv.org/abs/2506.00433

    2025 arXiv

  67. [75]

    Improving the diffusability of autoencoders

    Ivan Skorokhodov, Sharath Girish, Benran Hu, Willi Menapace, Yanyu Li, Rameen Abdal, Sergey Tulyakov, and Aliaksandr Siarohin. Improving the diffusability of autoencoders. InInt. Conf. Mach. Learn., 2025

    2025

  68. [76]

    Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole

    Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. InInt. Conf. Learn. Represent., 2021

    2021

  69. [77]

    Lssgen: Leveraging latent space scaling in flow and diffusion for efficient text to image generation

    Jyun-Ze Tang, Chih-Fan Hsu, Jeng-Lin Li, Ming-Ching Chang, and Wei-Chao Chen. Lssgen: Leveraging latent space scaling in flow and diffusion for efficient text to image generation. InInt. Conf. Comput. Vis., pages 5048–5057, 2025

    2025

  70. [79]

    Relay diffusion: Unifying diffusion process across resolutions for image synthesis

    Jiayan Teng, Wendi Zheng, Ming Ding, Wenyi Hong, Jianqiao Wangni, Zhuoyi Yang, and Jie Tang. Relay diffusion: Unifying diffusion process across resolutions for image synthesis. InInt. Conf. Learn. Represent.,

  71. [80]

    Visual autoregressive modeling: Scalable image generation via next-scale prediction

    Keyu Tian, Yi Jiang, Zehuan Yuan, Bingyue Peng, and Liwei Wang. Visual autoregressive modeling: Scalable image generation via next-scale prediction. InAdv. Neural Inform. Process. Syst., 2024

    2024

  72. [81]

    Training-free diffusion acceleration with bottleneck sampling, 2025

    Ye Tian, Xin Xia, Yuxi Ren, Shanchuan Lin, Xing Wang, Xuefeng Xiao, Yunhai Tong, Ling Yang, and Bin Cui. Training-free diffusion acceleration with bottleneck sampling, 2025. URL https://arxiv.org/ abs/2503.18940

    2025

  73. [82]

    Hiwave: Training-free high- resolution image generation via wavelet-based diffusion sampling

    Tobias V ontobel, Seyedmorteza Sadat, Farnood Salehi, and Romann Weber. Hiwave: Training-free high- resolution image generation via wavelet-based diffusion sampling. InProceedings of the SIGGRAPH Asia 2025 Conference Papers, pages 1–11, 2025. 14

    2025

  74. [83]

    Wan: Open and advanced large-scale video generative models, 2025

    Team Wan, Ang Wang, Baole Ai, Bin Wen, Chaojie Mao, Chen-Wei Xie, Di Chen, Feiwu Yu, Haiming Zhao, Jianxiao Yang, et al. Wan: Open and advanced large-scale video generative models, 2025. URL https://arxiv.org/abs/2503.20314

    2025 arXiv

  75. [84]

    Phased consistency models

    Fu-Yun Wang, Zhaoyang Huang, Alexander William Bergman, Dazhong Shen, Peng Gao, Michael Lingelbach, Keqiang Sun, Weikang Bian, Guanglu Song, Yu Liu, Hongsheng Li, and Hao Ouyang. Phased consistency models. InAdv. Neural Inform. Process. Syst., 2024

    2024

  76. [85]

    Jianyi Wang, Kelvin C. K. Chan, and Chen Change Loy. Exploring CLIP for assessing the look and feel of images. InAAAI, 2023

    2023

  77. [86]

    Linformer: Self-attention with linear complexity, 2020

    Sinong Wang, Belinda Z Li, Madian Khabsa, Hao Fang, and Hao Ma. Linformer: Self-attention with linear complexity, 2020. URLhttps://arxiv.org/abs/2006.04768

    2020 arXiv

  78. [87]

    Importance-based token merging for efficient image and video generation

    Haoyu Wu, Jingyi Xu, Hieu Le, and Dimitris Samaras. Importance-based token merging for efficient image and video generation. InInt. Conf. Comput. Vis., pages 4983–4995, 2025

    2025

  79. [88]

    Training-free and adaptive sparse attention for efficient long video generation

    Yifei Xia, Suhan Ling, Fangcheng Fu, Yujie Wang, Huixia Li, Xuefeng Xiao, and Bin Cui. Training-free and adaptive sparse attention for efficient long video generation. InInt. Conf. Comput. Vis., 2025

    2025

  80. [89]

    ImageReward: Learning and evaluating human preferences for text-to-image generation

    Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. ImageReward: Learning and evaluating human preferences for text-to-image generation. InAdv. Neural Inform. Process. Syst., 2023

    2023

  81. [90]

    CogVideoX: Text-to-video diffusion models with an expert transformer

    Zhuoyi Yang, Jiayan Teng, Wendi Zheng, Ming Ding, Shiyu Huang, Jiazheng Xu, Yuanming Yang, Wenyi Hong, Xiaohan Zhang, Guanyu Feng, et al. CogVideoX: Text-to-video diffusion models with an expert transformer. InInt. Conf. Learn. Represent., 2025

    2025

  82. [91]

    Tianwei Yin, Michaël Gharbi, Taesung Park, Richard Zhang, Eli Shechtman, Fredo Durand, and William T. Freeman. Improved distribution matching distillation for fast image synthesis. InAdv. Neural Inform. Process. Syst., 2024

    2024

  83. [92]

    Freeman, and Taesung Park

    Tianwei Yin, Michaël Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William T. Freeman, and Taesung Park. One-step diffusion with distribution matching distillation. InIEEE Conf. Comput. Vis. Pattern Recog., 2024

    2024

  84. [93]

    Freeman, Fredo Durand, Eli Shechtman, and Xide Xia

    Tianwei Yin, Qiang Zhang, Richard Zhang, William T. Freeman, Fredo Durand, Eli Shechtman, and Xide Xia. From slow bidirectional to fast autoregressive video diffusion models. InIEEE Conf. Comput. Vis. Pattern Recog., 2025

    2025

  85. [94]

    Bidirectional sparse attention for faster video diffusion training, 2025

    Chenlu Zhan, Wen Li, Chuyu Shen, Jun Zhang, Suhui Wu, and Hao Zhang. Bidirectional sparse attention for faster video diffusion training, 2025. URLhttps://arxiv.org/abs/2509.01085

    2025

  86. [95]

    Diffusion-4k: Ultra-high-resolution image synthesis with latent diffusion models

    Jinjin Zhang, Qiuyu Huang, Junjie Liu, Xiefan Guo, and Di Huang. Diffusion-4k: Ultra-high-resolution image synthesis with latent diffusion models. InIEEE Conf. Comput. Vis. Pattern Recog., 2025

    2025

  87. [96]

    From sketch to fresco: Efficient diffusion transformer with progressive resolution, 2026

    Shikang Zheng, Guantao Chen, Lixuan He, Jiacheng Liu, Yuqi Lin, Chang Zou, and Linfeng Zhang. From sketch to fresco: Efficient diffusion transformer with progressive resolution, 2026. URL https: //arxiv.org/abs/2601.07462

    2026

  88. [97]

    CogView3: Finer and faster text-to-image generation via relay diffusion

    Wendi Zheng, Jiayan Teng, Zhuoyi Yang, Weihan Wang, Jidong Chen, Xiaotao Gu, Yuxiao Dong, Ming Ding, and Jie Tang. CogView3: Finer and faster text-to-image generation via relay diffusion. InEur. Conf. Comput. Vis., 2024

    2024

  89. [98]

    Accelerating diffusion transformers with token-wise feature caching, 2024

    Chang Zou, Xuyang Liu, Ting Liu, Siteng Huang, and Linfeng Zhang. Accelerating diffusion transformers with token-wise feature caching, 2024. URLhttps://arxiv.org/abs/2410.05317. 15 Appendix Contents A Supporting Definitions and Proofs for Section 4.2 17 A.1 Definitions . . . ....

    2024

  90. [99]

    Setup.The spectral noise expansion produces an expanded state xsi+1 ti in the frequency domain of the expanded resolution grid. For a grid withN si+1 points, the coefficients are: xsi+1 ti (ω) = ( (1−t i) (xsi 0 )(ω) +t iϵ(ω), ω∈Ω si , tiϵ′(ω), ω∈Ω si+1 \Ω si , where ϵ(ω), ϵ′(...

  91. [100]

    By matching the noise coefficients across all bands and the signal coefficients on the shared band Ωsi, we derive two scalar conditions: κi ·t i = ˜ti, κ i(1−t i) =r(1− ˜ti)

    Timestep alignment.We seek a scaling factor κi and an aligned timestep ˜ti ∈(0,1) such that, for everyω∈Ω si+1, the transformed state satisfies: ˜xsi+1 ˜ti (ω) =κ i xsi+1 ti (ω) = (1− ˜ti) xsi+1 0 (ω) + ˜ti˜ϵ(ω),˜ϵ (ω) ∼ N(0,1)i.i.d. By matching the noise coefficients across a...

  92. [101]

    the signal and noise coefficients are independent

    Assumptions.The proof assumes a linear-Gaussian model, where: (A1) we model the clean data spectrum as zero-mean Gaussian x(ω) 0 ∼ N(0, P ω) with second momentP ω >0; (A2) we assumeϵ (ω) ∼ N(0,1), by orthonormality ofΦapplied toϵ∼ N(0, I); (A3) we assumex (ω) 0 ⊥ϵ (ω), i.e. th...

  93. [102]

    (7) and (A1)–(A3), E[x(ω) t ] = 0,Var(x (ω) t ) = (1−t) 2Pω +t 2,(20) Cov(x(ω) t , x (ω) 0 ) = (1−t)P ω,Cov(x (ω) t , ϵ (ω)) =t.(21)

    Second-moment quantities.From the per-frequency forward process in Eq. (7) and (A1)–(A3), E[x(ω) t ] = 0,Var(x (ω) t ) = (1−t) 2Pω +t 2,(20) Cov(x(ω) t , x (ω) 0 ) = (1−t)P ω,Cov(x (ω) t , ϵ (ω)) =t.(21)

  94. [103]

    Bayes-optimal velocity as a linear predictor.Under (A1)–(A3), the triplet (x(ω) 0 , ϵ (ω), x (ω) t ) is jointly Gaussian with zero mean, since x(ω) t = (1−t)x (ω) 0 +t ϵ (ω) is a linear combination of two independent zero-mean Gaussians. For any pair of zero-mean jointly Gauss...

  95. [104]

    (26), A= t−(1−t)P ω D .(30) Substituting into Eq

    Substitute and simplify.Plugging x(ω) t = (1−t)x (ω) 0 +t ϵ (ω) into (26) and collecting terms in x(ω) 0 andϵ (ω), v∗(ω) −ϵ (ω) =A (1−t)x (ω) 0 +t ϵ (ω) −ϵ (ω) =A(1−t)x (ω) 0 + (At−1)ϵ (ω).(27) Squaring and taking expectation, using (A1)–(A3) (so E[(x(ω) 0 )2] =P ω, E[(ϵ(ω))2]...

  96. [105]

    (15), the denominator is (1−t) 2Pω +t 2 =t 2(1 + SNRω(t)) and the numerator equals t2 SNRω(t) (1 +Pω)

    Rewrite via SNRω(t).Using SNRω(t) = (1−t) 2Pω/t2 from Eq. (15), the denominator is (1−t) 2Pω +t 2 =t 2(1 + SNRω(t)) and the numerator equals t2 SNRω(t) (1 +Pω). The t2 cancels, recovering the right-hand side of Eq. (19) and completing the proof of the lemma. Proposition 1(Per-...

  97. [106]

    Therefore the radicand δ/ Pω(1 +P ω −δ) in Eq

    tω is strictly increasing in Pω.For δ∈(0,1) and Pω >0 , the quadratic Pω (1 +P ω −δ) = P 2 ω + (1−δ)P ω has derivative 2Pω + (1−δ)>0 , hence is strictly increasing in Pω. Therefore the radicand δ/ Pω(1 +P ω −δ) in Eq. (9) is strictly decreasing in Pω, and so is its square root...

  98. [107]

    (4) gives Pω ∝ |ω| −β with β >0

    Pω is strictly decreasing in |ω|.The power-law assumption in Eq. (4) gives Pω ∝ |ω| −β with β >0. ThusP ω is strictly decreasing in|ω|

  99. [108]

    CHOOLBUS

    Minimum over Ωs.Combining the above observations, tω is strictly decreasing in |ω|. The set Ωs ={ω∈Ω :|ω| ≤s·ω max(H, W)} consists exactly of the frequencies with radius at most s·ω max(H, W), so min ω∈Ωs tω =t s·ωmax(H,W) ,(44) which is Eq. (10). 21 B Empirical Validation of ...

  100. [109]

    Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline

    matched to similar speedups. Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline. 32 Z-Image(1×)Z-Image(1.56×)Ours(1.65×)Z-Image(4.99×)Ours(5.04×)Ours* (5.01×) Figure 10:Qu...

  101. [110]

    Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline

    matched to similar speedups. Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline. 33 Z-Image(1×)Z-Image(1.56×)Ours(1.65×)Z-Image(4.99×)Ours(5.04×)Ours* (5.01×) Figure 11:Qu...

  102. [111]

    Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline

    matched to similar speedups. Our fine-tuned model (Ours∗) achieves even higher image quality compared to our training-free acceleration variant and outperforms the reduced-step baseline. 34 PixelGen(1×)PixelGen(1.9×)Ours(1.6×)Ours* (1.6×) Figure 12:Qualitative comparisons on p...

  103. [112]

    Compute the spectrum of the input imageξ in =T Φ(xin)

  104. [113]

    Extract the low-frequency part of ξin corresponding to the representable spectrum Ωsk at scales k

  105. [114]

    Fill the spectrumΩ sk+1 \Ω sk witht kϵ, whereϵ∼ N(0,1)

  106. [115]

    The total time of denoising is thus1−t k

    Convert the expanded spectrum Ωsk+1 back to the spatial domain xsk+1 tk via T −1 Φ and continue denoising starting from the transition timet k. The total time of denoising is thus1−t k. We conduct image editing experiments using Z-Image [5] as the backbone, evaluating our freq...

Pith tools

Reviewed May 21, 2026 · model on record in the stance chip above.