# Wavelet Score-Based Generative Modeling

arXiv: https://arxiv.org/abs/2208.05003 · HTML: https://mlsanity.com/paper/2208.05003/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 33 of its 65 references that are in the map.
- First posted on arXiv 9 Aug 2022.

## Placed in

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

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech](https://mlsanity.com/paper/2105.06337/) (2021)
- [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium](https://mlsanity.com/paper/1706.08500/) (2017)
- [On Fast Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.00132/) (2021)
- [A* Sampling](https://mlsanity.com/paper/1411.0030/) (2014)
- [Noise Estimation for Generative Diffusion Models](https://mlsanity.com/paper/2104.02600/) (2021)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)
- [Non Gaussian Denoising Diffusion Models](https://mlsanity.com/paper/2106.07582/) (2021)
- [Wavelet Flow: Fast Training of High Resolution Normalizing Flows](https://mlsanity.com/paper/2010.13821/) (2020)

## Built on by (most cited first, 30 of 62)

- [Diffusion Model-Based Image Editing: A Survey](https://mlsanity.com/paper/2402.17525/) (2024)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Wavelet Diffusion Models are fast and scalable Image Generators](https://mlsanity.com/paper/2211.16152/) (2022)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Long-Term Photometric Consistent Novel View Synthesis with Diffusion Models](https://mlsanity.com/paper/2304.10700/) (2023)
- [Wavelet Convolutions for Large Receptive Fields](https://mlsanity.com/paper/2407.05848/) (2024)
- [Diffusion-4K: Ultra-High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2503.18352/) (2025)
- [Diffusion Models, Image Super-Resolution And Everything: A Survey](https://mlsanity.com/paper/2401.00736/) (2024)
- [WDM: 3D Wavelet Diffusion Models for High-Resolution Medical Image Synthesis](https://mlsanity.com/paper/2402.19043/) (2024)
- [Boosting Diffusion Models with Moving Average Sampling in Frequency Domain](https://mlsanity.com/paper/2403.17870/) (2024)
- [A Fourier Space Perspective on Diffusion Models](https://mlsanity.com/paper/2505.11278/) (2025)
- [WaveDM: Wavelet-Based Diffusion Models for Image Restoration](https://mlsanity.com/paper/2305.13819/) (2023)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Fast Graph Generation via Spectral Diffusion](https://mlsanity.com/paper/2211.08892/) (2022)
- [Synthetic Lagrangian Turbulence by Generative Diffusion Models](https://mlsanity.com/paper/2307.08529/) (2023)
- [Accelerated Diffusion Models via Speculative Sampling](https://mlsanity.com/paper/2501.05370/) (2025)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [A Unified Framework for U-Net Design and Analysis](https://mlsanity.com/paper/2305.19638/) (2023)
- [Enhancing Foundation Models for Time Series Forecasting via Wavelet-based Tokenization](https://mlsanity.com/paper/2412.05244/) (2024)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [LiteVAE: Lightweight and Efficient Variational Autoencoders for Latent Diffusion Models](https://mlsanity.com/paper/2405.14477/) (2024)
- [Learning multi-scale local conditional probability models of images](https://mlsanity.com/paper/2303.02984/) (2023)
- [Waving Goodbye to Low-Res: A Diffusion-Wavelet Approach for Image Super-Resolution](https://mlsanity.com/paper/2304.01994/) (2023)
- [Generative modeling for time series via Schr"odinger bridge](https://mlsanity.com/paper/2304.05093/) (2023)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [Fr'echet Wavelet Distance: A Domain-Agnostic Metric for Image Generation](https://mlsanity.com/paper/2312.15289/) (2023)
- [Wavelets Are All You Need for Autoregressive Image Generation](https://mlsanity.com/paper/2406.19997/) (2024)
- [Image generation with shortest path diffusion](https://mlsanity.com/paper/2306.00501/) (2023)

## Abstract (arXiv)

Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. We show that SGMs can be considerably accelerated, by factorizing the data distribution into a product of conditional probabilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically over Gaussian distributions, and shown numerically over physical processes at phase transition and natural image datasets.