arXiv subjects

Hefin Lambley

Publications and source records attributed to Hefin Lambley.

4 recordsLinked to original sources

arXiv2025

Classification of small-ball modes and maximum a posteriori estimators in metric spaces

A mode, or `most likely point', for a probability measure $μ$ can be defined in various ways via the asymptotic behaviour of the $μ$-mass of balls as their radius tends to zero. Such points are of intrinsic interest in the local theory of measures on metric spaces and also arise naturally in the study of Bayesian inverse problems and diffusion processes. Building upon special cases already proposed in the literature, this paper develops a systematic framework for defining modes through small-ball probabilities. We propose `common-sense' axioms that such definitions should obey, including appropriate treatment of discrete and absolutely continuous measures, as well as symmetry and invariance properties. We show that there are exactly ten such definitions consistent with these axioms, and that they are partially but not totally ordered in strength, forming a complete, distributive lattice. We also show how this classification simplifies for well-behaved $μ$.

math.PR↗

arXiv2024

Autoencoders in Function Space

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.

stat.ML↗

arXiv2023

Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

This article shows that a large class of posterior measures that are absolutely continuous with respect to a Gaussian prior have strong maximum a posteriori estimators in the sense of Dashti et al. (2013). This result holds in any separable Banach space and applies in particular to nonparametric Bayesian inverse problems with additive noise. When applied to Bayesian inverse problems, this significantly extends existing results on maximum a posteriori estimators by relaxing the conditions on the log-likelihood and on the space in which the inverse problem is set.

math.ST↗

arXiv2023

An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems

It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of $X$, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of $X$, which we show can be dense in $X$, even for an absolutely continuous measure on $X = \mathbb{R}$.

math.ST↗