# Score-based Denoising Diffusion with Non-Isotropic Gaussian Noise Models

arXiv: https://arxiv.org/abs/2210.12254 · HTML: https://mlsanity.com/paper/2210.12254/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 16 of its 23 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Placed in

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

## Builds on (most cited first, 16 of 16)

- [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)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation](https://mlsanity.com/paper/2205.09853/) (2022)
- [Diffusion Probabilistic Modeling for Video Generation](https://mlsanity.com/paper/2203.09481/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [Adversarial score matching and improved sampling for image generation](https://mlsanity.com/paper/2009.05475/) (2020)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [Denoising Diffusion Gamma Models](https://mlsanity.com/paper/2110.05948/) (2021)

## Built on by (most cited first, 17 of 17)

- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [A Fourier Space Perspective on Diffusion Models](https://mlsanity.com/paper/2505.11278/) (2025)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Blue noise for diffusion models](https://mlsanity.com/paper/2402.04930/) (2024)
- [Memory in Plain Sight: Surveying the Uncanny Resemblances of Associative Memories and Diffusion Models](https://mlsanity.com/paper/2309.16750/) (2023)
- [Nonisotropic Gaussian Diffusion for Realistic 3D Human Motion Prediction](https://mlsanity.com/paper/2501.06035/) (2025)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [Edge-preserving noise for diffusion models](https://mlsanity.com/paper/2410.01540/) (2024)
- [Multi-Resolution Diffusion for Privacy-Sensitive Recommender Systems](https://mlsanity.com/paper/2311.03488/) (2023)
- [GUD: Generation with Unified Diffusion](https://mlsanity.com/paper/2410.02667/) (2024)
- [Learning What Matters: Steering Diffusion via Spectrally Anisotropic Forward Noise](https://mlsanity.com/paper/2510.09660/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Colorful-Noise: Training-Free Low-Frequency Noise Manipulation for Color-Based Conditional Image Generation](https://mlsanity.com/paper/2605.00548/) (2026)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Phi-Noise: Training-Free Temporal Video Conditioning via Phase-Based Noise Manipulation](https://mlsanity.com/paper/2605.24509/) (2026)
- [Noise in Diffusion Models Is a Learnable Input](https://mlsanity.com/paper/2608.02575/) (2026)

## Abstract (arXiv)

Generative models based on denoising diffusion techniques have led to an unprecedented increase in the quality and diversity of imagery that is now possible to create with neural generative models. However, most contemporary state-of-the-art methods are derived from a standard isotropic Gaussian formulation. In this work we examine the situation where non-isotropic Gaussian distributions are used. We present the key mathematical derivations for creating denoising diffusion models using an underlying non-isotropic Gaussian noise model. We also provide initial experiments with the CIFAR-10 dataset to help verify empirically that this more general modeling approach can also yield high-quality samples.