We consider the problem of recovering a function input of a differential equation formulated on an unknown domain $\mathcal{M}$. We assume to have access to a discrete domain $\mathcal{M}_n=\{\mathbf{x}_1, \dots, \mathbf{x}_n\} \subset \mathcal{M}$ 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 $\mathcal{M}_n$ converge, in the large $n$ limit, to a posterior over functions in $\mathcal{M}$ 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.


  1. graph Laplacian
  2. posterior
  3. continuum limits
  4. $\Gamma$-convergence

MSC codes

  1. 28A33
  2. 46N30
  3. 62F15

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4020 - 4040
ISSN (online): 1095-7154


Submitted: 10 July 2017
Accepted: 27 April 2018
Published online: 19 July 2018


  1. graph Laplacian
  2. posterior
  3. continuum limits
  4. $\Gamma$-convergence

MSC codes

  1. 28A33
  2. 46N30
  3. 62F15



Nicolás García Trillos

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