# Spectral Analysis of Diffusion Models with Application to Schedule Design

arXiv: https://arxiv.org/abs/2502.00180 · HTML: https://mlsanity.com/paper/2502.00180/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 34 of its 45 references that are in the map.
- First posted on arXiv 31 Jan 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [Fr'echet Audio Distance: A Metric for Evaluating Music Enhancement Algorithms](https://mlsanity.com/paper/1812.08466/) (2018)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [Scalable Adaptive Computation for Iterative Generation](https://mlsanity.com/paper/2212.11972/) (2022)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality](https://mlsanity.com/paper/2202.05830/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [DPM-Solver-v3: Improved Diffusion ODE Solver with Empirical Model Statistics](https://mlsanity.com/paper/2310.13268/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Intriguing properties of synthetic images: from generative adversarial networks to diffusion models](https://mlsanity.com/paper/2304.06408/) (2023)
- [Diffusion Probabilistic Model Made Slim](https://mlsanity.com/paper/2211.17106/) (2022)
- [Align Your Steps: Optimizing Sampling Schedules in Diffusion Models](https://mlsanity.com/paper/2404.14507/) (2024)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [On the Trajectory Regularity of ODE-based Diffusion Sampling](https://mlsanity.com/paper/2405.11326/) (2024)
- [Learning to Discretize Denoising Diffusion ODEs](https://mlsanity.com/paper/2405.15506/) (2024)
- [Time Series Diffusion in the Frequency Domain](https://mlsanity.com/paper/2402.05933/) (2024)

## Built on by (most cited first, 13 of 13)

- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Spectrally-Guided Diffusion Noise Schedules](https://mlsanity.com/paper/2603.19222/) (2026)
- [Analyzing and Guiding Zero-Shot Posterior Sampling in Diffusion Models](https://mlsanity.com/paper/2602.07715/) (2026)
- [Spectral Regularization for Diffusion Models](https://mlsanity.com/paper/2603.02447/) (2026)
- [A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials](https://mlsanity.com/paper/2603.18225/) (2026)
- [Mixture-of-Gaussians-Guided Schedule Design for Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2607.03517/) (2026)
- [SALD: Self-Referenced Advantage Learning for Diffusion Models](https://mlsanity.com/paper/2610.01496/) (2026)
- [A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Models](https://mlsanity.com/paper/2601.13602/) (2026)
- [Spectral Collapse in Diffusion Inversion](https://mlsanity.com/paper/2602.13303/) (2026)
- [Low-Pass Flow Matching](https://mlsanity.com/paper/2606.02177/) (2026)
- [PRISM: Principled Reference Identification for Schrodinger Bridge Model](https://mlsanity.com/paper/2608.06893/) (2026)
- [Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching](https://mlsanity.com/paper/2609.39488/) (2026)
- [Colored Graph Signal Diffusion Models](https://mlsanity.com/paper/2610.17282/) (2026)

## Abstract (arXiv)

Diffusion models (DMs) have emerged as powerful tools for modeling complex data distributions and generating realistic new samples. Over the years, advanced architectures and sampling methods have been developed to make these models practically usable. However, certain synthesis process decisions still rely on heuristics without a solid theoretical foundation. In our work, we offer a novel analysis of the DM's inference process, introducing a comprehensive frequency response perspective. Specifically, by relying on Gaussianity assumption, we present the inference process as a closed-form spectral transfer function, capturing how the generated signal evolves in response to the initial noise. We demonstrate how the proposed analysis can be leveraged to design a noise schedule that aligns effectively with the characteristics of the data. The spectral perspective also provides insights into the underlying dynamics and sheds light on the relationship between spectral properties and noise schedule structure. Our results lead to scheduling curves that are dependent on the spectral content of the data, offering a theoretical justification for some of the heuristics taken by practitioners.