# Improved DDIM Sampling with Moment Matching Gaussian Mixtures

arXiv: https://arxiv.org/abs/2311.04938 · HTML: https://mlsanity.com/paper/2311.04938/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 47 of its 67 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Placed in

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

## Builds on (most cited first, 30 of 47)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)

## Built on by (most cited first, 3 of 3)

- [Diffusion Time-step Curriculum for One Image to 3D Generation](https://mlsanity.com/paper/2404.04562/) (2024)
- [Universality of Gaussian-Mixture Reverse Kernels in Conditional Diffusion](https://mlsanity.com/paper/2604.13470/) (2026)
- [Learning Probabilistic Prompt for Continual Learning](https://mlsanity.com/paper/2607.04711/) (2026)

## Abstract (arXiv)

We propose using a Gaussian Mixture Model (GMM) as reverse transition operator (kernel) within the Denoising Diffusion Implicit Models (DDIM) framework, which is one of the most widely used approaches for accelerated sampling from pre-trained Denoising Diffusion Probabilistic Models (DDPM). Specifically we match the first and second order central moments of the DDPM forward marginals by constraining the parameters of the GMM. We see that moment matching is sufficient to obtain samples with equal or better quality than the original DDIM with Gaussian kernels. We provide experimental results with unconditional models trained on CelebAHQ and FFHQ, class-conditional models trained on ImageNet, and text-to-image generation using Stable Diffusion v2.1 on COYO700M datasets respectively. Our results suggest that using the GMM kernel leads to significant improvements in the quality of the generated samples when the number of sampling steps is small, as measured by FID and IS metrics. For example on ImageNet 256x256, using 10 sampling steps, we achieve a FID of 6.94 and IS of 207.85 with a GMM kernel compared to 10.15 and 196.73 respectively with a Gaussian kernel. Further, we derive novel SDE samplers for rectified flow matching models and experiment with the proposed approach. We see improvements using both 1-rectified flow and 2-rectified flow models. Code: https://github.com/pgabbur/ddim-gmm.