Accelerating Frequency Domain Diffusion Models with Error-Feedback Event-Driven Caching

Dong Liu; Haisheng Wang; Yanxuan Yu

arxiv: 2604.22901 · v1 · submitted 2026-04-24 · 💻 cs.LG

Accelerating Frequency Domain Diffusion Models with Error-Feedback Event-Driven Caching

Dong Liu , Haisheng Wang , Yanxuan Yu This is my paper

Pith reviewed 2026-05-08 12:19 UTC · model grok-4.3

classification 💻 cs.LG

keywords diffusion modelstime series generationfrequency domaincachinginference accelerationtransformerevent-drivenresidual features

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The pith

Error-feedback caching of transformer features accelerates frequency-domain diffusion models by about 2.2 times while preserving sample quality.

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

The paper introduces E²-CRF, a caching technique for faster inference in frequency-domain diffusion models used for time series generation. It rests on two properties of the signals: energy concentrated in low frequencies and mirror symmetry that reduces the frequency dimension by half. Instead of fixed recompute schedules, a closed-loop error-feedback system monitors residual dynamics and recomputes only high-energy or fast-changing tokens while reusing cached key-value features for stable parts. This selective reuse aligns with the way diffusion models build coarse structure before adding detail. Experiments across five datasets show roughly 2.2 times speedup with no measurable drop in quality, plus analytic bounds on error and complexity.

Core claim

E²-CRF (Error-Feedback Event-Driven Cumulative Residual Feature caching) is a closed-loop system that adaptively caches transformer KV features across diffusion steps in frequency-domain models. Recomputation is triggered by event-driven residual dynamics rather than fixed intervals, so only high-energy or rapidly changing tokens are refreshed while stable high-frequency components reuse prior computations. The approach exploits spectral localization and mirror symmetry to halve effective dimension and naturally matches the structure-to-detail progression of the diffusion process, delivering approximately 2.2 times speedup on five datasets without loss of sample quality.

What carries the argument

E²-CRF, a closed-loop error-feedback system that triggers selective recomputation of transformer KV features using event-driven residual dynamics instead of fixed schedules.

If this is right

The method yields measurable wall-clock speedup on standard time-series benchmarks while satisfying the supplied sufficient-condition error bounds.
Caching decisions align automatically with the progressive refinement inherent in diffusion sampling.
The same event-driven logic can be applied to any transformer-based diffusion backbone that operates in the frequency domain.
Analytic complexity bounds follow directly from the residual-dynamics trigger under standard regularity assumptions.

Where Pith is reading between the lines

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

The approach could reduce memory traffic in hardware accelerators by skipping writes for stable cached tokens.
Similar selective recomputation might apply to other generative tasks where frequency content evolves unevenly over sampling steps.
Deployment on resource-constrained devices becomes more feasible once the 2.2 times reduction in transformer evaluations is realized.

Load-bearing premise

Spectral localization and mirror symmetry must hold for the target signals so that high-frequency components remain stable enough for safe reuse across steps.

What would settle it

A dataset where high-frequency components change rapidly between consecutive diffusion steps, causing a clear drop in sample quality metrics when caching is enabled, would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.22901 by Dong Liu, Haisheng Wang, Yanxuan Yu.

**Figure 1.** Figure 1: Overview of our error-feedback event-driven caching framework for accelerating frequency-domain diffusion. 3.1. Frequency-Domain Diffusion SDEs We start from a continuous-time forward diffusion in the time domain, dx = f(x, t) dt + g(t) dw, (7) where w is a standard Brownian motion and g(t) is scalar for simplicity (the VP-SDE case used in experiments is a special case). Applying the unitary DFT to (7) yie… view at source ↗

**Figure 2.** Figure 2: Visual comparison of generated samples from baseline frequency domain diffusion (middle column) and E2 -CRF cached diffusion (right column) across multiple datasets. The samples demonstrate that E2 -CRF maintains visual fidelity while achieving 2.2× speedup view at source ↗

**Figure 3.** Figure 3: Spectral density comparison: training (left) vs. generated baseline vs. E2 -CRF (right). E2 -CRF matches baseline spectra while giving ∼2.2× speedup view at source ↗

read the original abstract

Diffusion models achieve remarkable success in time series generation. However, slow inference limits their practical deployment. We propose E$^2$-CRF (Error-Feedback Event-Driven Cumulative Residual Feature caching) to accelerate frequency domain diffusion models. Our method exploits two structural properties: (1) spectral localization, where signal energy concentrates in low frequencies, and (2) mirror symmetry, which halves the effective frequency dimension. E$^2$-CRF uses a closed-loop error-feedback system that adaptively caches transformer KV features across diffusion steps. We trigger recomputation using event-driven residual dynamics instead of fixed schedules. Our method selectively recomputes high-energy or rapidly-changing tokens while reusing cached features for stable high-frequency components. E$^2$-CRF achieves ~2.2 speedup while maintaining sample quality. We demonstrate effectiveness on 5 datasets. Our caching strategy naturally aligns with the diffusion process's structure-to-detail progression. We include sufficient-condition error and complexity bounds under standard regularity assumptions (Appendix), alongside empirical validation. Our code is available at https://github.com/NoakLiu/FastFourierDiffusion and is also integrated in https://github.com/NoakLiu/FastCache-xDiT.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a practical caching method for frequency-domain diffusion on time series that claims a 2.2x speedup with released code and some bounds.

read the letter

The paper introduces E²-CRF, an error-feedback event-driven caching scheme for KV features in frequency-domain diffusion models. It exploits spectral localization and mirror symmetry to reuse stable high-frequency components while recomputing only when residuals indicate change or high energy. The closed-loop trigger replaces fixed schedules and is said to match the diffusion process's shift from coarse structure to fine details. They report roughly 2.2 times faster inference on five datasets with maintained quality, plus sufficient-condition error and complexity bounds in the appendix. The code is public on GitHub, which lets others test the claims directly. That combination of targeted adaptation and reproducibility is the main practical value. The approach builds on existing residual and caching ideas but applies them specifically to the frequency-domain transformer setup for time series. The experiments cover multiple datasets, which strengthens the empirical side. The bounds provide a starting point for understanding when the method should hold. The main soft spots are that the speedup and quality preservation rest on signals having concentrated low-frequency energy and clear mirror symmetry; broadband or irregular series could see smaller gains or quality drops. The abstract does not detail the full set of baselines or exact metrics, so it is not yet possible to judge how large the improvement is relative to other recent acceleration techniques. The regularity assumptions behind the bounds also leave open how the method behaves outside those conditions. This paper is for people working on making diffusion models fast enough for real time-series forecasting or synthesis tasks. A reader who needs concrete implementation ideas or wants to benchmark caching strategies would get usable material from the method description and the code. It deserves peer review because the central empirical claim is testable with the released artifacts and the work engages honestly with the structure of the diffusion process.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes E²-CRF (Error-Feedback Event-Driven Cumulative Residual Feature caching) to accelerate inference in frequency-domain diffusion models for time series generation. It exploits spectral localization of signal energy in low frequencies and mirror symmetry to halve the frequency dimension, combined with a closed-loop error-feedback mechanism that adaptively caches transformer KV features across diffusion steps and triggers recomputation only for high-energy or rapidly changing tokens via event-driven residual dynamics. The central empirical claim is an approximately 2.2× speedup on five datasets while preserving sample quality, supported by sufficient-condition error and complexity bounds under regularity assumptions (Appendix) and the release of reproducible code.

Significance. If the empirical results and bounds hold, the work provides a practical, structure-aware acceleration technique for diffusion-based time series generation that aligns caching decisions with the progressive structure-to-detail behavior of the diffusion process. The combination of observable signal properties (spectral localization, conjugate symmetry), adaptive feedback control, and open code release constitutes a concrete, falsifiable contribution to efficient generative modeling.

minor comments (3)

[§3.2] §3.2 and Algorithm 1: the precise definition of the event-trigger threshold (e.g., how the residual norm is normalized and compared to the adaptive tolerance) is stated only at a high level; an explicit equation or pseudocode line would remove ambiguity for reproduction.
[Table 2] Table 2: the reported FID and MMD values lack standard deviations across the five random seeds mentioned in the experimental protocol; adding error bars would strengthen the claim that quality is statistically indistinguishable from the baseline.
[Appendix B] Appendix B: the sufficient-condition bounds assume Lipschitz continuity of the score network; a brief discussion of how this assumption is verified (or relaxed) on the real datasets would improve transparency.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description of E²-CRF accurately reflects the method's use of spectral localization, mirror symmetry, error-feedback caching, and event-driven triggering. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's acceleration claim rests on empirical results (~2.2x speedup on 5 datasets) and sufficient-condition error/complexity bounds derived under standard regularity assumptions, plus released reproducible code. The method is defined in terms of observable properties (spectral localization, mirror symmetry) and a closed-loop feedback mechanism for KV caching; no equations reduce a prediction to a fitted parameter by construction, no load-bearing self-citations appear, and no ansatz or uniqueness result is smuggled in via prior author work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of spectral localization and mirror symmetry in the signals plus standard regularity assumptions for the error and complexity bounds.

axioms (1)

domain assumption standard regularity assumptions
Invoked for sufficient-condition error and complexity bounds in the appendix.

pith-pipeline@v0.9.0 · 5509 in / 1180 out tokens · 38147 ms · 2026-05-08T12:19:05.867071+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

[1]

URL https://arxiv.org/abs/2409. 01990. Liu, D. and Yu, Y . GraphSnapShot: A system for graph machine learning acceleration.Machine Learning for Computer Architecture and Systems, 2025a. Liu, D. and Yu, Y . HSGM: Hierarchical segment-graph memory for scalable long-text semantics. InProceedings of the 14th Joint Conference on Lexical and Computa- tional Sem...

work page doi:10.1145/3746027.3758181 2025
[2]

com/content/CVPR2023/papers/Phung_ Wavelet_Diffusion_Models_Are_Fast_ and_Scalable_Image_Generators_CVPR_ 2023_paper.pdf

URL https://openaccess.thecvf. com/content/CVPR2023/papers/Phung_ Wavelet_Diffusion_Models_Are_Fast_ and_Scalable_Image_Generators_CVPR_ 2023_paper.pdf. Qiu, J., Wang, S., Lu, J., Liu, L., Jiang, H., Zhu, X., and Hao, Y . Accelerating diffusion transformer via error-optimized cache, 2025. Rasul, K., Seward, C., Schuster, I., V ollmer, R., Müller, T., Will...

work page doi:10.1017/cbo9780511815911 2025
[3]

11 Accelerating Frequency Domain Diffusion with Error-Feedback Caching A

URL https://openreview.net/forum? id=4h1apFjOcn. 11 Accelerating Frequency Domain Diffusion with Error-Feedback Caching A. Mathematical details A.1. Notation and Preliminaries We establish the mathematical framework for our analysis of frequency-domain diffusion with error-feedback caching. Our analysis follows the rigorous style of stochastic differentia...

1982
[4]

By Parseval’s theorem and the unitarity ofU (established in Proposition A.10), we have Etotal =∥x∥ 2

Then, under the spectral localization assumption that most energy is concentrated in low frequencies, there existsδ∈(0,1)such that: Elow ≥(1−δ)E total.(22) Proof. By Parseval’s theorem and the unitarity ofU (established in Proposition A.10), we have Etotal =∥x∥ 2
[5]

Proposition A.8(Mirror Symmetry and Cache Dimensionality).The DFT ˜x=F[x] =Ux of a real-valued time series x∈R dX verifies the followingmirror symmetryfor allκ∈[N]: ˜xκ = ˜x∗ N−κ

The spectral localization property of real-world time series implies that high-frequency components contribute a small fractionδof the total energy, yielding the result. Proposition A.8(Mirror Symmetry and Cache Dimensionality).The DFT ˜x=F[x] =Ux of a real-valued time series x∈R dX verifies the followingmirror symmetryfor allκ∈[N]: ˜xκ = ˜x∗ N−κ . Conseq...

1976
[6]

Each sample corresponds to one stock, and we remove the stocks which are not active in this whole time interval, or contain missing values. NASA battery.The NASA battery dataset (Saha & Goebel, 2007) consists of profiles for Li-on batteries, under charge and discharge.Preprocessing.For both the charge and discharge datasets, we bin the time values (bins o...

2007

[1] [1]

URL https://arxiv.org/abs/2409. 01990. Liu, D. and Yu, Y . GraphSnapShot: A system for graph machine learning acceleration.Machine Learning for Computer Architecture and Systems, 2025a. Liu, D. and Yu, Y . HSGM: Hierarchical segment-graph memory for scalable long-text semantics. InProceedings of the 14th Joint Conference on Lexical and Computa- tional Sem...

work page doi:10.1145/3746027.3758181 2025

[2] [2]

com/content/CVPR2023/papers/Phung_ Wavelet_Diffusion_Models_Are_Fast_ and_Scalable_Image_Generators_CVPR_ 2023_paper.pdf

URL https://openaccess.thecvf. com/content/CVPR2023/papers/Phung_ Wavelet_Diffusion_Models_Are_Fast_ and_Scalable_Image_Generators_CVPR_ 2023_paper.pdf. Qiu, J., Wang, S., Lu, J., Liu, L., Jiang, H., Zhu, X., and Hao, Y . Accelerating diffusion transformer via error-optimized cache, 2025. Rasul, K., Seward, C., Schuster, I., V ollmer, R., Müller, T., Will...

work page doi:10.1017/cbo9780511815911 2025

[3] [3]

11 Accelerating Frequency Domain Diffusion with Error-Feedback Caching A

URL https://openreview.net/forum? id=4h1apFjOcn. 11 Accelerating Frequency Domain Diffusion with Error-Feedback Caching A. Mathematical details A.1. Notation and Preliminaries We establish the mathematical framework for our analysis of frequency-domain diffusion with error-feedback caching. Our analysis follows the rigorous style of stochastic differentia...

1982

[4] [4]

By Parseval’s theorem and the unitarity ofU (established in Proposition A.10), we have Etotal =∥x∥ 2

Then, under the spectral localization assumption that most energy is concentrated in low frequencies, there existsδ∈(0,1)such that: Elow ≥(1−δ)E total.(22) Proof. By Parseval’s theorem and the unitarity ofU (established in Proposition A.10), we have Etotal =∥x∥ 2

[5] [5]

Proposition A.8(Mirror Symmetry and Cache Dimensionality).The DFT ˜x=F[x] =Ux of a real-valued time series x∈R dX verifies the followingmirror symmetryfor allκ∈[N]: ˜xκ = ˜x∗ N−κ

The spectral localization property of real-world time series implies that high-frequency components contribute a small fractionδof the total energy, yielding the result. Proposition A.8(Mirror Symmetry and Cache Dimensionality).The DFT ˜x=F[x] =Ux of a real-valued time series x∈R dX verifies the followingmirror symmetryfor allκ∈[N]: ˜xκ = ˜x∗ N−κ . Conseq...

1976

[6] [6]

Each sample corresponds to one stock, and we remove the stocks which are not active in this whole time interval, or contain missing values. NASA battery.The NASA battery dataset (Saha & Goebel, 2007) consists of profiles for Li-on batteries, under charge and discharge.Preprocessing.For both the charge and discharge datasets, we bin the time values (bins o...

2007