# Conditioning non-linear and infinite-dimensional diffusion processes

arXiv: https://arxiv.org/abs/2402.01434 · HTML: https://mlsanity.com/paper/2402.01434/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 2 Feb 2024.

## Placed in

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

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Grounding DINO: Marrying DINO with Grounded Pre-Training for Open-Set Object Detection](https://mlsanity.com/paper/2303.05499/) (2023)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [Continuous-Time Functional Diffusion Processes](https://mlsanity.com/paper/2303.00800/) (2023)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States](https://mlsanity.com/paper/2303.18242/) (2023)
- [Diffusion bridges for stochastic Hamiltonian systems and shape evolutions](https://mlsanity.com/paper/2002.00885/) (2020)
- [Automatic Backward Filtering Forward Guiding for Markov processes and graphical models](https://mlsanity.com/paper/2010.03509/) (2020)
- [A Stochastic Large Deformation Model for Computational Anatomy](https://mlsanity.com/paper/1612.05323/) (2016)
- [A Geometric Framework for Stochastic Shape Analysis](https://mlsanity.com/paper/1703.09971/) (2017)

## Built on by (most cited first, 15 of 15)

- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Conditioning Diffusions Using Malliavin Calculus](https://mlsanity.com/paper/2504.03461/) (2025)
- [Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series](https://mlsanity.com/paper/2410.05602/) (2024)
- [Measurement Score-Based Diffusion Model](https://mlsanity.com/paper/2505.11853/) (2025)
- [Infinite-dimensional Diffusion Bridge Simulation via Operator Learning](https://mlsanity.com/paper/2405.18353/) (2024)
- [Neural Guided Diffusion Bridges](https://mlsanity.com/paper/2502.11909/) (2025)
- [SciPostGen: Bridging the Gap between Scientific Papers and Poster Layouts](https://mlsanity.com/paper/2511.22490/) (2025)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [Diffusion differentiable resampling](https://mlsanity.com/paper/2512.10401/) (2025)
- [Stochastics of shapes and Kunita flows](https://mlsanity.com/paper/2512.11676/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference](https://mlsanity.com/paper/2604.20775/) (2026)
- [Stochastic Interpolants in Hilbert Spaces](https://mlsanity.com/paper/2602.01988/) (2026)
- [Infinite-dimensional generative diffusions via Doob's h-transform](https://mlsanity.com/paper/2602.06621/) (2026)
- [Conditioning Degenerate Diffusion Models](https://mlsanity.com/paper/2609.04090/) (2026)

## Abstract (arXiv)

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.