# Generative Fractional Diffusion Models

arXiv: https://arxiv.org/abs/2310.17638 · HTML: https://mlsanity.com/paper/2310.17638/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 38 of its 79 references that are in the map.
- First posted on arXiv 26 Oct 2023.

## Placed in

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

## Builds on (most cited first, 30 of 38)

- [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)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Simple and Controllable Music Generation](https://mlsanity.com/paper/2306.05284/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [AudioGen: Textually Guided Audio Generation](https://mlsanity.com/paper/2209.15352/) (2022)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Soft Diffusion: Score Matching for General Corruptions](https://mlsanity.com/paper/2209.05442/) (2022)
- [Blurring Diffusion Models](https://mlsanity.com/paper/2209.05557/) (2022)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Dirichlet Diffusion Score Model for Biological Sequence Generation](https://mlsanity.com/paper/2305.10699/) (2023)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Maximum Likelihood Training of Implicit Nonlinear Diffusion Models](https://mlsanity.com/paper/2205.13699/) (2022)

## Built on by (most cited first, 12 of 12)

- [From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training](https://mlsanity.com/paper/2501.06148/) (2025)
- [Heavy-Tailed Diffusion with Denoising L'evy Probabilistic Models](https://mlsanity.com/paper/2407.18609/) (2024)
- [DMLR: Data-centric Machine Learning Research -- Past, Present and Future](https://mlsanity.com/paper/2311.13028/) (2023)
- [Generative modelling with jump-diffusions](https://mlsanity.com/paper/2503.06558/) (2025)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)
- [ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space](https://mlsanity.com/paper/2604.27443/) (2026)
- [Score Change of Variables](https://mlsanity.com/paper/2412.07904/) (2024)
- [Improving Diffusion Generative Models via Truncated Karhunen--Lo`eve Expansion](https://mlsanity.com/paper/2503.17657/) (2025)
- [Variational Inference for L'evy Process-Driven SDEs via Neural Tilting](https://mlsanity.com/paper/2605.10934/) (2026)
- [LongSpike: Fractional Order Spiking State Space Models for Efficient Long Sequence Learning](https://mlsanity.com/paper/2606.12895/) (2026)
- [Volterra Generative Models](https://mlsanity.com/paper/2606.18071/) (2026)
- [Fractional State Space Transition for Long Sequence Modeling](https://mlsanity.com/paper/2609.36314/) (2026)

## Abstract (arXiv)

We introduce the first continuous-time score-based generative model that leverages fractional diffusion processes for its underlying dynamics. Although diffusion models have excelled at capturing data distributions, they still suffer from various limitations such as slow convergence, mode-collapse on imbalanced data, and lack of diversity. These issues are partially linked to the use of light-tailed Brownian motion (BM) with independent increments. In this paper, we replace BM with an approximation of its non-Markovian counterpart, fractional Brownian motion (fBM), characterized by correlated increments and Hurst index H ∈ (0,1), where H=0.5 recovers the classical BM. To ensure tractable inference and learning, we employ a recently popularized Markov approximation of fBM (MA-fBM) and derive its reverse-time model, resulting in generative fractional diffusion models (GFDM). We characterize the forward dynamics using a continuous reparameterization trick and propose augmented score matching to efficiently learn the score function, which is partly known in closed form, at minimal added cost. The ability to drive our diffusion model via MA-fBM offers flexibility and control. H ≤ 0.5 enters the regime of rough paths whereas H>0.5 regularizes diffusion paths and invokes long-term memory. The Markov approximation allows added control by varying the number of Markov processes linearly combined to approximate fBM. Our evaluations on real image datasets demonstrate that GFDM achieves greater pixel-wise diversity and enhanced image quality, as indicated by a lower FID, offering a promising alternative to traditional diffusion models