# DDIM Redux: Mathematical Foundation and Some Extension

arXiv: https://arxiv.org/abs/2408.07285 · HTML: https://mlsanity.com/paper/2408.07285/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 3 Aug 2024.

## Placed in

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

## Builds on (most cited first, 9 of 9)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [gDDIM: Generalized denoising diffusion implicit models](https://mlsanity.com/paper/2206.05564/) (2022)
- [Fluctuation-dissipation Type Theorem in Stochastic Linear Learning](https://mlsanity.com/paper/2106.02220/) (2021)

## Built on by (most cited first, 1 of 1)

- [NatADiff: Adversarial Boundary Guidance for Natural Adversarial Diffusion](https://mlsanity.com/paper/2505.20934/) (2025)

## Abstract (arXiv)

This note provides a critical review of the mathematical concepts underlying the generalized diffusion denoising implicit model (gDDIM) and the exponential integrator (EI) scheme. We present enhanced mathematical results, including an exact expression for the reverse trajectory in the probability flow ODE and an exact expression for the covariance matrix in the gDDIM scheme. Furthermore, we offer an improved understanding of the EI scheme's efficiency in terms of the change of variables. The noising process in DDIM is analyzed from the perspective of non-equilibrium statistical physics. Additionally, we propose a new scheme for DDIM, called the principal-axis DDIM (paDDIM).