#publication
Transporting QMC points in Statistics and Computing
The article “Transporting higher-order quadrature rules - Quasi-Monte Carlo points and sparse grids for mixture distributions” by Ilja Klebanov and myself has just been published in its final form in Statistics and Computing.
I. Klebanov and T. J. Sullivan. “Transporting higher-order quadrature rules: Quasi-Monte Carlo points and sparse grids for mixture distributions.” Statistics and Computing 36(1):no. 46, 19pp., 2026.
Abstract. Integration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention in recent years has been the generation of samples from a target distribution \(\mathbb{P}_{\mathrm{tar}}\) using transport maps: if \(\mathbb{P}_{\mathrm{tar}} = T_{\sharp} \mathbb{P}_{\mathrm{ref}}\) is the pushforward of an easily-sampled probability distribution \(\mathbb{P}_{\mathrm{ref}}\) under the transport map \(T\), then the application of \(T\) to \(\mathbb{P}_{\mathrm{ref}}\)-distributed samples yields \(\mathbb{P}_{\mathrm{tar}}\)-distributed samples. This paper proposes the application of transport maps not just to random samples, but also to quasi-Monte Carlo points, higher-order nets, and sparse grids so that the transformed samples inherit the original convergence rates that are often better than \(N^{-1/2}\), \(N\) being the number of samples/quadrature nodes. Our main result is the derivation of an explicit transport map for the case that \(\mathbb{P}_{\mathrm{tar}}\) is a mixture of simple distributions, e.g. a Gaussian mixture, in which case application of the transport map \(T\) requires the solution of an explicit ODE with closed-form right-hand side. Mixture distributions are of particular applicability and interest since many methods proceed by first approximating \(\mathbb{P}_{\mathrm{tar}}\) by a mixture and then sampling from that mixture (often using importance reweighting). Hence, this paper allows for the sampling step to provide a better convergence rate than \(N^{-1/2}\) for all such methods.
Published on Tuesday 23 December 2025 at 12:00 UTC #publication #stat-comput #qmc #klebanov
Autoencoders in function space in JMLR
The article “Autoencoders in function space” by Justin Bunker, Mark Girolami, Hefin Lambley, Andrew Stuart and myself has just appeared in its final form in the Journal of Machine Learning Research. This article continues one of the main themes of my and collaborators' work, namely that powerful discretisation-invariant learning methods can be obtained by examining the problem in an infinite-dimensional function space instead of on a fixed grid.
Abstract. Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.
Published on Sunday 7 September 2025 at 12:00 UTC #publication #jmlr #bunker #girolami #lambley #stuart #autoencoders
Hille's theorem for locally convex spaces in Real Analysis Exchange
The article “Hille's theorem for Bochner integrals of functions with values in locally convex spaces” has just appeared in its final form in Real Analysis Exchange.
T. J. Sullivan. “Hille's theorem for Bochner integrals of functions with values in locally convex spaces.” Real Analysis Exchange 49(2):377–388, 2024.
Abstract. Hille's theorem is a powerful classical result in vector measure theory. It asserts that the application of a closed, unbounded linear operator commutes with strong/Bochner integration of functions taking values in a Banach space. This note shows that Hille's theorem also holds in the setting of complete locally convex spaces.
Published on Tuesday 1 October 2024 at 13:00 UTC #publication #real-anal-exch #functional-analysis #hille-theorem
Unbounded images of Gaussian and other stochastic processes in Analysis and Applications
The final version of “Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators” by Tadashi Matsumoto and myself has just appeared in Analysis and Applications.
The purpose of this article is to provide a self-contained rigorous proof of the well-known formula for the mean and covariance function of a stochastic process — in particular, a Gaussian process — when it is acted upon by an unbounded linear operator such as an ordinary or partial differential operator, as used in probabilistic approaches to the solution of ODEs and PDEs. This result is easy to establish in the case of a bounded operator, but the unbounded case requires a careful application of Hille's theorem for the Bochner integral of a Banach-valued random variable.
T. Matsumoto and T. J. Sullivan. “Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators.” Analysis and Applications 22(3):619–633, 2024.
Abstract. Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP \(v\) that is the image of another GP \(u\) under a linear transformation \(T\) acting on the sample paths of \(u\) are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when \(T\) is an unbounded operator such as a differential operator, which is common in several modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator \(T\) acting on the sample paths of a square-integrable stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable.
Published on Wednesday 21 February 2024 at 10:00 UTC #publication #anal-appl #prob-num #gp #matsumoto
Order-theoretic perspectives on MAP estimation in SIAM/ASA JUQ
The final version of “An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems” by Hefin Lambley and myself has just appeared online in the SIAM/ASA Journal on Uncertainty Quantification.
On a heuristic level, modes and MAP estimators are intended to be the “most probable points” of a space \(X\) with respect to a probability measure \(\mu\). Thus, in some sense, they would seem to be the greatest elements of some order on \(X\), and a rigorous order-theoretic treatment is called for, especially for cases in which \(X\) is, say, an infinite-dimensional function space. Such an order-theoretic perspective opens up some attractive proof strategies for the existence of modes and MAP estimators but also leads to some interesting counterexamples. In particular, because the orders involved are not total, some pairs of points of \(X\) can be incomparable (i.e. neither is more nor less likely than the other). In fact we show that there are examples for which the collection of such mutually incomparable elements is dense in \(X\).
H. Lambley and T. J. Sullivan. “An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems.” SIAM/ASA Journal on Uncertainty Quantification 11(4):1195–1224, 2023.
Published on Friday 20 October 2023 at 09:00 UTC #publication #modes #order-theory #map-estimators #lambley #juq