arXiv subjects

Michael Hutchinson

Publications and source records attributed to Michael Hutchinson.

12 recordsLinked to original sources

arXiv2026

Spectral Diffusion Processes

Diffusion models have proven to be a flexible and effective framework for modelling probability distributions on finite-dimensional spaces. However, many physical modelling problems such as time series are naturally described over function spaces. In this work we apply diffusion models to such stochastic processes. To do so we consider a spectral representation of the data, obtained using a kernel, thereby dissociating the stochastic part of the processes from their space-time structure. As a result, the stochasticity of the processes is entirely encoded in the spectral coefficients, which we truncate and model using standard finite-dimensional diffusion models. By truncating the representation in the spectral domain we ensure our resulting model defines valid stochastic processes, thereby naturally satisfying consistency and exchangeability criteria. Projecting our spectral diffusion models back to the original input space, we show that for any given marginals our approach corresponds to a diffusion model with correlated noise, with explicit covariance matrix given by the kernel. We demonstrate our method's effectiveness for modelling various multimodal datasets as well as conditional sampling by amortising our models with respect to a context set.

stat.ML↗

arXiv2024

Diffusion Models for Constrained Domains

Denoising diffusion models are a novel class of generative algorithms that achieve state-of-the-art performance across a range of domains, including image generation and text-to-image tasks. Building on this success, diffusion models have recently been extended to the Riemannian manifold setting, broadening their applicability to a range of problems from the natural and engineering sciences. However, these Riemannian diffusion models are built on the assumption that their forward and backward processes are well-defined for all times, preventing them from being applied to an important set of tasks that consider manifolds defined via a set of inequality constraints. In this work, we introduce a principled framework to bridge this gap. We present two distinct noising processes based on (i) the logarithmic barrier metric and (ii) the reflected Brownian motion induced by the constraints. As existing diffusion model techniques cannot be applied in this setting, we derive new tools to define such models in our framework. We then demonstrate the practical utility of our methods on a number of synthetic and real-world tasks, including applications from robotics and protein design.

cs.LG↗

arXiv2024

Target Score Matching

Denoising Score Matching estimates the score of a noised version of a target distribution by minimizing a regression loss and is widely used to train the popular class of Denoising Diffusion Models. A well known limitation of Denoising Score Matching, however, is that it yields poor estimates of the score at low noise levels. This issue is particularly unfavourable for problems in the physical sciences and for Monte Carlo sampling tasks for which the score of the clean original target is known. Intuitively, estimating the score of a slightly noised version of the target should be a simple task in such cases. In this paper, we address this shortcoming and show that it is indeed possible to leverage knowledge of the target score. We present a Target Score Identity and corresponding Target Score Matching regression loss which allows us to obtain score estimates admitting favourable properties at low noise levels.

cs.LG↗

arXiv2023

Metropolis Sampling for Constrained Diffusion Models

Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains. In addition, their extension to Riemannian manifolds has facilitated a range of applications across the natural sciences. While many of these problems stand to benefit from the ability to specify arbitrary, domain-informed constraints, this setting is not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by constructing novel noising processes based on the reflected Brownian motion and logarithmic barrier methods. However, the associated samplers are either computationally burdensome or only apply to convex subsets of Euclidean space. In this paper, we introduce an alternative, simple noising scheme based on Metropolis sampling that affords substantial gains in computational efficiency and empirical performance compared to the earlier samplers. Of independent interest, we prove that this new process corresponds to a valid discretisation of the reflected Brownian motion. We demonstrate the scalability and flexibility of our approach on a range of problem settings with convex and non-convex constraints, including applications from geospatial modelling, robotics and protein design.

cs.LG↗

arXiv2022

Riemannian Score-Based Generative Modelling

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

cs.LG↗

arXiv2022

Riemannian Diffusion Schrödinger Bridge

Score-based generative models exhibit state of the art performance on density estimation and generative modeling tasks. These models typically assume that the data geometry is flat, yet recent extensions have been developed to synthesize data living on Riemannian manifolds. Existing methods to accelerate sampling of diffusion models are typically not applicable in the Riemannian setting and Riemannian score-based methods have not yet been adapted to the important task of interpolation of datasets. To overcome these issues, we introduce \emph{Riemannian Diffusion Schrödinger Bridge}. Our proposed method generalizes Diffusion Schrödinger Bridge introduced in \cite{debortoli2021neurips} to the non-Euclidean setting and extends Riemannian score-based models beyond the first time reversal. We validate our proposed method on synthetic data and real Earth and climate data.

stat.ML↗

arXiv2021

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.

stat.ML↗

arXiv2021

Equivariant Learning of Stochastic Fields: Gaussian Processes and Steerable Conditional Neural Processes

Motivated by objects such as electric fields or fluid streams, we study the problem of learning stochastic fields, i.e. stochastic processes whose samples are fields like those occurring in physics and engineering. Considering general transformations such as rotations and reflections, we show that spatial invariance of stochastic fields requires an inference model to be equivariant. Leveraging recent advances from the equivariance literature, we study equivariance in two classes of models. Firstly, we fully characterise equivariant Gaussian processes. Secondly, we introduce Steerable Conditional Neural Processes (SteerCNPs), a new, fully equivariant member of the Neural Process family. In experiments with Gaussian process vector fields, images, and real-world weather data, we observe that SteerCNPs significantly improve the performance of previous models and equivariance leads to improvements in transfer learning tasks.

cs.LG↗

arXiv2021

LieTransformer: Equivariant self-attention for Lie Groups

Group equivariant neural networks are used as building blocks of group invariant neural networks, which have been shown to improve generalisation performance and data efficiency through principled parameter sharing. Such works have mostly focused on group equivariant convolutions, building on the result that group equivariant linear maps are necessarily convolutions. In this work, we extend the scope of the literature to self-attention, that is emerging as a prominent building block of deep learning models. We propose the LieTransformer, an architecture composed of LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups. We demonstrate the generality of our approach by showing experimental results that are competitive to baseline methods on a wide range of tasks: shape counting on point clouds, molecular property regression and modelling particle trajectories under Hamiltonian dynamics.

cs.LG↗

arXiv2021

Enhanced Direct Delta Mush

Direct Delta Mush is a novel skinning deformation technique introduced by Le and Lewis (2019). It generalizes the iterative Delta Mush algorithm of Mancewicz et al (2014), providing a direct solution with improved efficiency and control. Compared to Linear Blend Skinning, Direct Delta Mush offers better quality of deformations and ease of authoring at comparable performance. However, Direct Delta Mush does not handle non-rigid joint transformations correctly which limits its application for most production environments. This paper presents an extension to Direct Delta Mush that integrates the non-rigid part of joint transformations into the algorithm. In addition, the paper also describes practical considerations for computing the orthogonal component of the transformation and stability issues observed during the implementation and testing.

cs.GR↗

arXiv2019

Differentially Private Federated Variational Inference

In many real-world applications of machine learning, data are distributed across many clients and cannot leave the devices they are stored on. Furthermore, each client's data, computational resources and communication constraints may be very different. This setting is known as federated learning, in which privacy is a key concern. Differential privacy is commonly used to provide mathematical privacy guarantees. This work, to the best of our knowledge, is the first to consider federated, differentially private, Bayesian learning. We build on Partitioned Variational Inference (PVI) which was recently developed to support approximate Bayesian inference in the federated setting. We modify the client-side optimisation of PVI to provide an ($ε$, $δ$)-DP guarantee. We show that it is possible to learn moderately private logistic regression models in the federated setting that achieve similar performance to models trained non-privately on centralised data.

stat.ML↗

arXiv2018

Ultrafast Magnetic Resonance Imaging

We propose a radical advance in Magnetic Resonance Imaging. MRI remains slow because it requires successive applications of magnetic field gradients to encode for spatial location. Parallel MRI accelerates imaging by permitting undersampling of k-space using multiple detectors, each with a unique spatial sensitivity to the radio field emitted by the object. This is called spatial sensitivity encoding. A fourfold undersampling means a fourfold acceleration, however a five-fold undersampling cannot be reconstructed. This is because too much reliance is placed on spatial sensitivity encoding, and spatial sensitivity encoding is less efficient than gradient encoding when the detectors are large. On the contrary, we hypothesized that when very large numbers of very small detectors are deployed, it would render spatial sensitivity encoding efficient, and would allow complete elimination of gradient reversals and therefore extremely rapid MR imaging. Since each detector receives all of the signal from all of the object all of the time, signal is expected to be high, and each detector is predicted to receive signal of the same order of magnitude as one large detector in conventional MRI. Moreover, the elimination of gradients removes the major source of electrical noise. Unexpectedly, therefore, we anticipate that signal-to-noise (SNR) will be high. By means of a detailed simulation, in which the exact Hertzian radio field is simulated for each of 32x32x32 voxels interacting with each of 1024 detector loops, we demonstrate that 3-dimensional MR images can indeed be acquired very quickly. Nevertheless, we also simulated imaging in the presence of unforeseen sources of noise, and found that image reconstruction is resistant to noise, even with unrealistically low SNR values. In summary, high quality volumetric MRI is likely to be achievable in milliseconds.

physics.med-ph↗