Jakiw Pidstrigach

Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich et al.

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modelling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with current canonical choices. For other distributions, however, we can improve upon these canonical choices. We demonstrate these results both theoretically and empirically, by applying the algorithms to data distributions on manifolds and to distributions arising in Bayesian inverse problems or simulation-based inference.

Jakiw Pidstrigach

Score-based generative models (SGMs) need to approximate the scores $\nabla \log p_t$ of the intermediate distributions as well as the final distribution $p_T$ of the forward process. The theoretical underpinnings of the effects of these approximations are still lacking. We find precise conditions under which SGMs are able to produce samples from an underlying (low-dimensional) data manifold $\mathcal{M}$. This assures us that SGMs are able to generate the "right kind of samples". For example, taking $\mathcal{M}$ to be the subset of images of faces, we find conditions under which the SGM robustly produces an image of a face, even though the relative frequencies of these images might not accurately represent the true data generating distribution. Moreover, this analysis is a first step towards understanding the generalization properties of SGMs: Taking $\mathcal{M}$ to be the set of all training samples, our results provide a precise description of when the SGM memorizes its training data.

Alex Dobra, Jakiw Pidstrigach, Tim Reichelt et al.

Sensitivity analysis is a cornerstone of climate science, essential for understanding phenomena ranging from storm intensity to long-term climate feedbacks. However, computing these sensitivities using traditional physical models is often prohibitively expensive in terms of both computation and development time. While modern AI-based generative models are orders of magnitude faster to evaluate, computing sensitivities with them remains a significant bottleneck. This work addresses this challenge by applying the adjoint state method for calculating gradients in generative flow models. We apply this method to the cBottle generative model, trained on ERA5 and ICON data, to perform sensitivity analysis of any atmospheric variable with respect to sea surface temperatures. We quantitatively validate the computed sensitivities against the model's own outputs. Our results provide initial evidence that this approach can produce reliable gradients, reducing the computational cost of sensitivity analysis from weeks on a supercomputer with a physical model to hours on a GPU, thereby simplifying a critical workflow in climate science. The code can be found at https://github.com/Kwartzl8/cbottle_adjoint_sensitivity.

Samuel Howard, Nikolas Nüsken, Jakiw Pidstrigach

Simulating the conditioned dynamics of diffusion processes, given their initial and terminal states, is an important but challenging problem in the sciences. The difficulty is particularly pronounced for rare events, for which the unconditioned dynamics rarely reach the terminal state. In this work, we leverage a self-consistency property of the conditioned dynamics to learn the diffusion bridge in an iterative online manner, and demonstrate promising empirical results in a range of settings.

Diksha Bhandari, Jakiw Pidstrigach, Sebastian Reich

We consider the problem of performing Bayesian inference for logistic regression using appropriate extensions of the ensemble Kalman filter. Two interacting particle systems are proposed that sample from an approximate posterior and prove quantitative convergence rates of these interacting particle systems to their mean-field limit as the number of particles tends to infinity. Furthermore, we apply these techniques and examine their effectiveness as methods of Bayesian approximation for quantifying predictive uncertainty in neural networks.

Andreas Bergmeister, Stefanie Jegelka, Nikolas Nüsken et al.

Diffusion and flow-matching models scale because pretraining is supervised regression: a clean sample is noised analytically, and a model regresses against a closed-form target. RL post-training aligns the model with a reward. In image generation, this makes samples compose objects correctly, render text legibly, and match human preferences. Existing methods rely on costly SDE rollouts, reward gradients, or surrogate losses, sacrificing pretraining's regression structure. We show that the structure extends to RL post-training. Under KL-regularized reward maximization, the optimal generative process tilts the clean-endpoint distribution towards samples with higher reward and leaves the noising law unchanged. Combining this with the adjoint-matching optimality condition and a REINFORCE identity, we derive Reinforce Adjoint Matching (RAM): a consistency loss that corrects the pretraining target with the reward. At each step, we draw a clean endpoint from the current model, evaluate its reward, noise it as in pretraining, and regress. No SDE rollouts, backward adjoint sweeps, or reward gradients are required. Like the pretraining objective, RAM is simple and scales. On Stable Diffusion 3.5M, RAM achieves the highest reward on composability, text rendering, and human preference, reaching Flow-GRPO's peak reward in up to $50\times$ fewer training steps.

Jakiw Pidstrigach, Elizabeth Baker, Carles Domingo-Enrich et al. · oxford

In generative modelling and stochastic optimal control, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and centred around a generalisation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singularities. This allows our approach to handle a broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. As a byproduct, we also introduce a novel score matching objective. Our loss functions are formulated such that they could readily be extended to manifold-valued and infinite dimensional diffusions.

Tyler Farghly, Peter Potaptchik, Samuel Howard et al.

Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function -- or equivalently, smoothing in the log-density domain -- produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.

Jakiw Pidstrigach

In this article, we consider the preconditioned Hamiltonian Monte Carlo (pHMC) algorithm defined directly on an infinite-dimensional Hilbert space. In this context, and under a condition reminiscent of strong log-concavity of the target measure, we prove convergence bounds for adjusted pHMC in the standard 1-Wasserstein distance. The arguments rely on a synchronous coupling of two copies of pHMC, which is controlled by adapting elements from arXiv:1805.00452.