Peter Potaptchik
University of Oxford
peter.potaptchik@stats.ox.ac.uk
Harvard University
peter_potaptchik@seas.harvard.edu
I'm a DPhil student at the University of Oxford, advised by Yee Whye Teh, and currently based at Harvard University as a Fellow, where I work with Michael Albergo.
My doctoral research is supported by a Google PhD Fellowship, an NSERC Postgraduate Scholarship, and the EPSRC StatML CDT.
Previously, I worked with Chris J. Maddison and Daniel Roy at the University of Toronto, where I completed my BSc in Computer Science and Statistics.
Feel free to reach out if you'd like to collaborate or just chat!
News
2025
- Awarded a Google PhD Fellowship
- Started as a Fellow at Harvard University
- Won first prize at the Citadel Research Showcase
- Awarded an NSERC Postgraduate Scholarship
2023
- Won the short talk award at BAYSM (j-ISBA)
- Started my DPhil at the University of Oxford
Research
My research focuses on diffusion and flow matching models, which frame generative modelling as transporting noise to data through the simulation of differential equations. These dynamical approaches unlock a powerful set of tools for alignment during both training and inference, capabilities that are central to modern large-scale generative models.
More recently, I have been interested in few-step generation for diffusion and flow-based models. Because these methods typically rely on expensive iterative simulation, there is growing interest in approaches such as consistency models and flow maps, which compress long rollouts into much cheaper generation procedures. Beyond improving efficiency, these approaches also open up new opportunities for alignment and adaptation.
Looking ahead, I am particularly interested in bringing these ideas to language, where flow-based methods may offer a path not only to cheaper and faster generation than standard autoregressive models, but also to stronger and more efficient post-training techniques.
Papers
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Discrete Flow Maps
The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative integration. Flow Maps bypass this bottleneck by compressing generative trajectories into single-step mappings, theoretically enabling the generation of full text sequences from noise in a single forward pass. However, standard formulations rely on Euclidean regression losses that are geometrically ill-suited for discrete data. In this work, we resolve this conflict with Discrete Flow Maps, a framework that reconciles trajectory compression with the geometry of the probability simplex. We recast standard flow map training for the discrete domain, aligning the training dynamics with the discrete nature of language. Empirically, this strict geometric alignment allows our method to surpass previous state-of-the-art results in discrete flow modeling.
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Meta Flow Maps enable scalable reward alignment
Controlling generative models—whether via inference-time steering or fine-tuning—is expensive. Control relies on estimating the value function—typically necessitating costly trajectory simulations. To eliminate this bottleneck, we introduce Meta Flow Maps (MFMs), stochastic extensions of consistency models and flow maps. MFMs are trained to perform one-step posterior sampling, generating arbitrarily many i.i.d. draws of clean data \(x_1\) from any noisy state \(x_t\). Crucially, these samples are differentiable in the conditioning state \(x_t\), unlocking efficient estimation of the value function gradient. We leverage this capability to enable both inference-time steering without inner rollouts, and unbiased, off-policy fine-tuning to general rewards. Among our fine-tuning and steering experiments on ImageNet, we highlight that our single-particle steered-MFM sampler outperforms a Best-of-1000 baseline across multiple rewards at a fraction of the compute.
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Tilt Matching for Scalable Sampling and Fine-Tuning
We propose a simple, scalable algorithm for using stochastic interpolants to sample from unnormalized densities and for fine-tuning generative models. The approach, Tilt Matching, arises from a dynamical equation relating the flow matching velocity to one targeting the same distribution tilted by a reward, implicitly solving a stochastic optimal control problem. The new velocity inherits the regularity of stochastic interpolant transports while also being the minimizer of an objective with strictly lower variance than flow matching itself. The update to the velocity field can be interpreted as the sum of all joint cumulants of the stochastic interpolant and copies of the reward, and to first order is their covariance. The algorithms do not require any access to gradients of the reward or backpropagating through trajectories of the flow or diffusion. We empirically verify that the approach is efficient and highly scalable, providing state-of-the-art results on sampling under Lennard-Jones potentials and is competitive on fine-tuning Stable Diffusion, without requiring reward multipliers. It can also be straightforwardly applied to tilting few-step flow map models.
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CREPE: Controlling Diffusion with Replica Exchange
Inference-time control of diffusion models aims to steer model outputs to satisfy new constraints without retraining. Previous approaches have mostly relied on heuristic guidance or have been coupled with Sequential Monte Carlo (SMC) for bias correction. In this paper, we propose a flexible alternative based on replica exchange, an algorithm designed initially for sampling problems. We refer to this method as CREPE (Controlling with REPlica Exchange). Unlike SMC, CREPE: (1) generates particles sequentially, (2) maintains high diversity in the generated samples after a burn-in period, and (3) enables online refinement or early termination. We demonstrate its versatility across various tasks, including temperature annealing, reward-tilting, model composition and classifier-free guidance debiasing, with competitive performance compared to prior SMC methods.
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Accelerated Parallel Tempering via Neural Transports
Markov Chain Monte Carlo (MCMC) algorithms are essential tools in computational statistics for sampling from unnormalised probability distributions, but can be fragile when targeting high-dimensional, multimodal, or complex target distributions. Parallel Tempering (PT) enhances MCMC's sample efficiency through annealing and parallel computation, propagating samples from tractable reference distributions to intractable targets via state swapping across interpolating distributions. The effectiveness of PT is limited by the often minimal overlap between adjacent distributions in challenging problems, which requires increasing the computational resources to compensate. We introduce a framework that accelerates PT by leveraging neural samplers -- including normalising flows, diffusion models, and controlled diffusions -- to reduce the required overlap. Our approach utilises neural samplers in parallel, circumventing the computational burden of neural samplers while preserving the asymptotic consistency of classical PT. We demonstrate theoretically and empirically on a variety of multimodal sampling problems that our method improves sample quality, reduces the computational cost compared to classical PT, and enables efficient free energy/normalising constant estimation.
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Adaptive Diffusion Guidance via Stochastic Optimal Control
Guidance is a cornerstone of modern diffusion models, playing a pivotal role in conditional generation and enhancing the quality of unconditional samples. However, current approaches to guidance scheduling--determining the appropriate guidance weight--are largely heuristic and lack a solid theoretical foundation. This work addresses these limitations on two fronts. First, we provide a theoretical formalization that precisely characterizes the relationship between guidance strength and classifier confidence. Second, building on this insight, we introduce a stochastic optimal control framework that casts guidance scheduling as an adaptive optimization problem. In this formulation, guidance strength is not fixed but dynamically selected based on time, the current sample, and the conditioning class, either independently or in combination. By solving the resulting control problem, we establish a principled foundation for more effective guidance in diffusion models.
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Linear Convergence of Diffusion Models Under the Manifold Hypothesis
Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower \(d\)-dimensional manifold embedded into \(D\)-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in \(D\) or polynomial (superlinear) in \(d\). The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler (KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension \(d\). Moreover, we show that this linear dependency is sharp.
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Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive
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.
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Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres
Recent advances in flow-based generative modelling have provided scalable methods for computing the Schrödinger Bridge (SB) between distributions, a dynamic form of entropy-regularised Optimal Transport (OT) for the quadratic cost. The successful Iterative Markovian Fitting (IMF) procedure solves the SB problem via sequential bridge-matching steps, presenting an elegant and practical approach with many favourable properties over the more traditional Iterative Proportional Fitting (IPF) procedure. Beyond the standard setting, optimal transport can be generalised to the multi-marginal case in which the objective is to minimise a cost defined over several marginal distributions. Of particular importance are costs defined over a tree structure, from which Wasserstein barycentres can be recovered as a special case. In this work, we extend the IMF procedure to solve for the tree-structured SB problem. Our resulting algorithm inherits the many advantages of IMF over IPF approaches in the tree-based setting. In the case of Wasserstein barycentres, our approach can be viewed as extending the widely used fixed-point approach to use flow-based entropic OT solvers, while requiring only simple bridge-matching steps at each iteration.
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Metric Flow Matching for Smooth Interpolations on the Data Manifold
Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of Euclidean geometry, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.
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de Finetti's theorem and the existence of regular conditional distributions and strong laws on exchangeable algebras
We show the following generalizations of the de Finetti–Hewitt–Savage theorem: Given an exchangeable sequence of random elements, the sequence is conditionally i.i.d. if and only if each random element admits a regular conditional distribution given the exchangeable σ-algebra (equivalently, the shift invariant or the tail algebra). We use this result, which holds without any regularity or technical conditions, to demonstrate that any exchangeable sequence of random elements whose common distribution is Radon is conditional iid.