O. Deniz Akyildiz

Alex Glyn-Davies, Arnaud Vadeboncoeur, O. Deniz Akyildiz et al.

Variational inference (VI) is a computationally efficient and scalable methodology for approximate Bayesian inference. It strikes a balance between accuracy of uncertainty quantification and practical tractability. It excels at generative modelling and inversion tasks due to its built-in Bayesian regularisation and flexibility, essential qualities for physics related problems. For such problems, the underlying physical model determines the dependence between variables of interest, which in turn will require a tailored derivation for the central VI learning objective. Furthermore, in many physical inference applications this structure has rich meaning and is essential for accurately capturing the dynamics of interest. In this paper, we provide an accessible and thorough technical introduction to VI for forward and inverse problems, guiding the reader through standard derivations of the VI framework and how it can best be realized through deep learning. We then review and unify recent literature exemplifying the flexibility allowed by VI. This paper is designed for a general scientific audience looking to solve physics-based problems with an emphasis on uncertainty quantification

James Matthew Young, O. Deniz Akyildiz

With the advent of diffusion models, new proteins can be generated at an unprecedented rate. The motif scaffolding problem requires steering this generative process to yield proteins with a desirable functional substructure called a motif. While models have been trained to take the motif as conditional input, recent techniques in diffusion posterior sampling can be leveraged as zero-shot alternatives whose approximations can be corrected with sequential Monte Carlo (SMC) algorithms. In this work, we introduce a new set of guidance potentials for describing scaffolding tasks and solve them by adapting SMC-aided diffusion posterior samplers with an unconditional model, Genie, as a prior. In single motif problems, we find that (i) the proposed potentials perform comparably, if not better, than the conventional masking approach, (ii) samplers based on reconstruction guidance outperform their replacement method counterparts, and (iii) measurement tilted proposals and twisted targets improve performance substantially. Furthermore, as a demonstration, we provide solutions to two multi-motif problems by pairing reconstruction guidance with an SE(3)-invariant potential. We also produce designable internally symmetric monomers with a guidance potential for point symmetry constraints. Our code is available at: https://github.com/matsagad/mres-project.

James Matthew Young, Paula Cordero-Encinar, Sebastian Reich et al.

We develop a diffusion-based sampler for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a (simple) base distribution and the target distribution, widely used in diffusion models. Our approach is based on a practical implementation of diffusion-annealed Langevin Monte Carlo, which approximates the diffusion path with convergence guarantees. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions. We further develop novel control variate schedules that minimise the variance of these score estimates. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets.

James Cuin, Davide Carbone, Yanbo Tang et al.

The problem of optimising functions with intractable gradients frequently arise in machine learning and statistics, ranging from maximum marginal likelihood estimation procedures to fine-tuning of generative models. Stochastic approximation methods for this class of problems typically require inner sampling loops to obtain (biased) stochastic gradient estimates, which rapidly becomes computationally expensive. In this work, we develop sequential Monte Carlo (SMC) samplers for optimisation of functions with intractable gradients. Our approach replaces expensive inner sampling methods with efficient SMC approximations, which can result in significant computational gains. We establish convergence results for the basic recursions defined by our methodology which SMC samplers approximate. We demonstrate the effectiveness of our approach on the reward-tuning of energy-based models within various settings.

Alessio Spagnoletti, Tim Y. J. Wang, Marcelo Pereyra et al.

Vision-Language Latent Diffusion Models (LDMs) (Rombach et al., 2022) provide powerful generative priors for inverse problems. However, existing LDM-based inverse solvers typically require a large number of neural function evaluations (NFEs) and backpropagation through large pretrained components, leading to substantial computational costs and, in some cases, degraded reconstruction quality. We propose a unified Euclidean-Wasserstein-2 gradient-flow framework that jointly performs posterior sampling and prompt optimization in the latent space through a single flow that aligns the prior and posterior with the observed data. Combined with few-step latent text-to-image models, this formulation enables low-NFE inference without backpropagation through autoencoders. Experiments across several canonical imaging inverse problems show that our method achieves state-of-the-art performance with significantly reduced computational cost.

Paula Cordero-Encinar, O. Deniz Akyildiz, Andrew B. Duncan

We investigate the theoretical properties of general diffusion (interpolation) paths and their Langevin Monte Carlo implementation, referred to as diffusion annealed Langevin Monte Carlo (DALMC), under weak conditions on the data distribution. Specifically, we analyse and provide non-asymptotic error bounds for the annealed Langevin dynamics where the path of distributions is defined as Gaussian convolutions of the data distribution as in diffusion models. We then extend our results to recently proposed heavy-tailed (Student's t) diffusion paths, demonstrating their theoretical properties for heavy-tailed data distributions for the first time. Our analysis provides theoretical guarantees for a class of score-based generative models that interpolate between a simple distribution (Gaussian or Student's t) and the data distribution in finite time. This approach offers a broader perspective compared to standard score-based diffusion approaches, which are typically based on a forward Ornstein-Uhlenbeck (OU) noising process.

Paul Felix Valsecchi Oliva, O. Deniz Akyildiz, Andrew Duncan

We propose a continuous-time formulation of persistent contrastive divergence (PCD) for maximum likelihood estimation (MLE) of unnormalised densities. Our approach expresses PCD as a coupled, multiscale system of stochastic differential equations (SDEs), which perform optimisation of the parameter and sampling of the associated parametrised density, simultaneously. From this novel formulation, we are able to derive explicit bounds for the error between the PCD iterates and the MLE solution for the model parameter. This is made possible by deriving uniform-in-time (UiT) bounds for the difference in moments between the multiscale system and the averaged regime. An efficient implementation of the continuous-time scheme is introduced, leveraging a class of explicit, stable intregators, stochastic orthogonal Runge-Kutta Chebyshev (S-ROCK), for which we provide explicit error estimates in the long-time regime. This leads to a novel method for training energy-based models (EBMs) with explicit error guarantees.

Tim Y. J. Wang, Juan Kuntz, O. Deniz Akyildiz

We introduce a novel particle-based algorithm for end-to-end training of latent diffusion models. We reformulate the training task as minimizing a free energy functional and obtain a gradient flow that does so. By approximating the latter with a system of interacting particles, we obtain the algorithm, which we underpin theoretically by providing error guarantees. The novel algorithm compares favorably in experiments with previous particle-based methods and variational inference analogues.

O. Deniz Akyildiz, Pierre Del Moral, Joaquín Miguez

Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finite-dimensional solutions cannot be computed. This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges.

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we demonstrate the empirical effectiveness of our method on synthetic and image datasets.

Tim Y. J. Wang, O. Deniz Akyildiz

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a regularized Wasserstein gradient flow of the Kullback-Leibler divergence in the latent space. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.