# Continuum Limit of Posteriors in Graph Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/1706.07193 · HTML: https://mlsanity.com/paper/1706.07193/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2017.

## Built on by (most cited first, 4 of 4)

- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Gradient Flow Sampler-based Distributionally Robust Optimization](https://mlsanity.com/paper/2510.25956/) (2025)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)

## Abstract (arXiv)

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 Mn=x1, dots, xn subset M, and to noisy measurements of the output solution at p≤ n of those points. We introduce a graph-based Bayesian inverse problem, and show that the graph-posterior measures over functions in Mn 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.