Continuum Limit of Posteriors in Graph Bayesian Inverse Problems
This work provides foundational uncertainty quantification for graph-based machine learning tasks, such as inverse problems on unknown manifolds, but is incremental as it builds on existing Bayesian and graph methods.
The paper tackles the problem of recovering a function input of a differential equation on an unknown domain using noisy measurements from a discrete point cloud, showing that graph-based Bayesian posteriors converge to a continuum posterior in the large data limit, with proofs relying on variational formulations and new topologies for measure convergence.
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.