# f-DM: A Multi-stage Diffusion Model via Progressive Signal Transformation

arXiv: https://arxiv.org/abs/2210.04955 · HTML: https://mlsanity.com/paper/2210.04955/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 41 of its 48 references that are in the map.
- First posted on arXiv 10 Oct 2022.

## Placed in

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

## Builds on (most cited first, 30 of 41)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (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)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [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)

## Built on by (most cited first, 30 of 32)

- [Simple diffusion: End-to-end diffusion for high resolution images](https://mlsanity.com/paper/2301.11093/) (2023)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [TRACT: Denoising Diffusion Models with Transitive Closure Time-Distillation](https://mlsanity.com/paper/2303.04248/) (2023)
- [Snap Video: Scaled Spatiotemporal Transformers for Text-to-Video Synthesis](https://mlsanity.com/paper/2402.14797/) (2024)
- [BOOT: Data-free Distillation of Denoising Diffusion Models with Bootstrapping](https://mlsanity.com/paper/2306.05544/) (2023)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Simpler Diffusion (SiD2): 1.5 FID on ImageNet512 with pixel-space diffusion](https://mlsanity.com/paper/2410.19324/) (2024)
- [Matryoshka Diffusion Models](https://mlsanity.com/paper/2310.15111/) (2023)
- [Helios: Real Real-Time Long Video Generation Model](https://mlsanity.com/paper/2603.04379/) (2026)
- [Hume: Introducing System-2 Thinking in Visual-Language-Action Model](https://mlsanity.com/paper/2505.21432/) (2025)
- [Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling](https://mlsanity.com/paper/2404.12940/) (2024)
- [VolumeDiffusion: Flexible Text-to-3D Generation with Efficient Volumetric Encoder](https://mlsanity.com/paper/2312.11459/) (2023)
- [Schedule Your Edit: A Simple yet Effective Diffusion Noise Schedule for Image Editing](https://mlsanity.com/paper/2410.18756/) (2024)
- [RodinHD: High-Fidelity 3D Avatar Generation with Diffusion Models](https://mlsanity.com/paper/2407.06938/) (2024)
- [DART: Denoising Autoregressive Transformer for Scalable Text-to-Image Generation](https://mlsanity.com/paper/2410.08159/) (2024)
- [Control3Diff: Learning Controllable 3D Diffusion Models from Single-view Images](https://mlsanity.com/paper/2304.06700/) (2023)
- [Hierarchical Patch Diffusion Models for High-Resolution Video Generation](https://mlsanity.com/paper/2406.07792/) (2024)
- [Neural Diffusion Models](https://mlsanity.com/paper/2310.08337/) (2023)
- [FreCaS: Efficient Higher-Resolution Image Generation via Frequency-aware Cascaded Sampling](https://mlsanity.com/paper/2410.18410/) (2024)
- [Scale-wise Distillation of Diffusion Models](https://mlsanity.com/paper/2503.16397/) (2025)
- [StraIT: Non-autoregressive Generation with Stratified Image Transformer](https://mlsanity.com/paper/2303.00750/) (2023)
- [GIViC: Generative Implicit Video Compression](https://mlsanity.com/paper/2503.19604/) (2025)
- [Improving Progressive Generation with Decomposable Flow Matching](https://mlsanity.com/paper/2506.19839/) (2025)
- [Sequential Data Generation with Groupwise Diffusion Process](https://mlsanity.com/paper/2310.01400/) (2023)
- [Simplified Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2403.14623/) (2024)
- [Frequency-Domain Refinement with Multiscale Diffusion for Super Resolution](https://mlsanity.com/paper/2405.10014/) (2024)
- [Incorporating Pre-trained Diffusion Models in Solving the Schr"odinger Bridge Problem](https://mlsanity.com/paper/2508.18095/) (2025)
- [LowDiff: Efficient Diffusion Sampling with Low-Resolution Condition](https://mlsanity.com/paper/2509.15342/) (2025)
- [Data-Efficient Brushstroke Generation with Diffusion Models for Oil Painting](https://mlsanity.com/paper/2603.01103/) (2026)

## Abstract (arXiv)

Diffusion models (DMs) have recently emerged as SoTA tools for generative modeling in various domains. Standard DMs can be viewed as an instantiation of hierarchical variational autoencoders (VAEs) where the latent variables are inferred from input-centered Gaussian distributions with fixed scales and variances. Unlike VAEs, this formulation limits DMs from changing the latent spaces and learning abstract representations. In this work, we propose f-DM, a generalized family of DMs which allows progressive signal transformation. More precisely, we extend DMs to incorporate a set of (hand-designed or learned) transformations, where the transformed input is the mean of each diffusion step. We propose a generalized formulation and derive the corresponding de-noising objective with a modified sampling algorithm. As a demonstration, we apply f-DM in image generation tasks with a range of functions, including down-sampling, blurring, and learned transformations based on the encoder of pretrained VAEs. In addition, we identify the importance of adjusting the noise levels whenever the signal is sub-sampled and propose a simple rescaling recipe. f-DM can produce high-quality samples on standard image generation benchmarks like FFHQ, AFHQ, LSUN, and ImageNet with better efficiency and semantic interpretation.