Continuum Limit of Posteriors in Graph Bayesian Inverse Problems

TL;DR

This paper establishes the convergence of graph-based Bayesian inverse solutions to their continuum counterparts as the number of data points increases, providing a foundation for uncertainty quantification in graph-based machine learning tasks.

Contribution

It introduces a novel framework for analyzing the continuum limit of graph-based Bayesian inverse problems, including a new topology for measure convergence on point clouds.

Findings

Graph-posterior measures converge to continuum posteriors as data points grow.

The framework applies to inverse problems on unknown manifolds.

Provides theoretical basis for robust uncertainty quantification in graph-based learning.

Abstract

We consider the problem of recovering a function input of a differential equation formulated on an unknown domain $M$. We assume to have access to a discrete domain $M_n=\{x_1, \dots, x_n\} \subset M$, and to noisy measurements of the output solution at $p\le n$ of those points. We introduce a graph-based Bayesian inverse problem, and show that the graph-posterior measures over functions in $M_n$ converge, in the large $n$ limit, to a posterior over functions in $M$ that solves a Bayesian inverse problem with known domain. The proofs rely on the variational formulation of the Bayesian update, and on a new topology for the study of convergence of measures over functions on point clouds to a measure over functions on the continuum. Our framework, techniques, and results may serve to lay the foundations of robust uncertainty quantification of graph-based tasks in machine learning. The ideas are presented in the concrete setting of recovering the initial condition of the heat equation on an unknown manifold.