# Diffusion Models with Deterministic Normalizing Flow Priors

arXiv: https://arxiv.org/abs/2309.01274 · HTML: https://mlsanity.com/paper/2309.01274/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 55 of its 65 references that are in the map.
- First posted on arXiv 3 Sep 2023.

## Placed in

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

## Builds on (most cited first, 30 of 55)

- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [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)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [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)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation](https://mlsanity.com/paper/2205.09853/) (2022)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)

## Built on by (most cited first, 7 of 7)

- [Efficient Diffusion Models: A Survey](https://mlsanity.com/paper/2502.06805/) (2025)
- [On the Challenges and Opportunities in Generative AI](https://mlsanity.com/paper/2403.00025/) (2024)
- [REGLUE Your Latents with Global and Local Semantics for Entangled Diffusion](https://mlsanity.com/paper/2512.16636/) (2025)
- [The Coupling Within: Flow Matching via Distilled Normalizing Flows](https://mlsanity.com/paper/2603.09014/) (2026)
- [Survey on Visual Signal Coding and Processing with Generative Models: Technologies, Standards and Optimization](https://mlsanity.com/paper/2405.14221/) (2024)
- [Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transport](https://mlsanity.com/paper/2410.13431/) (2024)
- [Spatially-Grounded Flow Matching: Structured Source Distributions for Image Generation](https://mlsanity.com/paper/2608.15452/) (2026)

## Abstract (arXiv)

For faster sampling and higher sample quality, we propose DiNof (Diffusion with Normalizing flow priors), a technique that makes use of normalizing flows and diffusion models. We use normalizing flows to parameterize the noisy data at any arbitrary step of the diffusion process and utilize it as the prior in the reverse diffusion process. More specifically, the forward noising process turns a data distribution into partially noisy data, which are subsequently transformed into a Gaussian distribution by a nonlinear process. The backward denoising procedure begins with a prior created by sampling from the Gaussian distribution and applying the invertible normalizing flow transformations deterministically. To generate the data distribution, the prior then undergoes the remaining diffusion stochastic denoising procedure. Through the reduction of the number of total diffusion steps, we are able to speed up both the forward and backward processes. More importantly, we improve the expressive power of diffusion models by employing both deterministic and stochastic mappings. Experiments on standard image generation datasets demonstrate the advantage of the proposed method over existing approaches. On the unconditional CIFAR10 dataset, for example, we achieve an FID of 2.01 and an Inception score of 9.96. Our method also demonstrates competitive performance on CelebA-HQ-256 dataset as it obtains an FID score of 7.11. Code is available at https://github.com/MohsenZand/DiNof.