Continuum Limits of Posteriors in Graph Bayesian Inverse Problems
Continuum Limits of Posteriors in Graph Bayesian Inverse Problems
复制标题
图贝叶斯反问题中后验的连续统极限
DOI：
10.1137/17m1138005
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
D. Sanz
中科院分区：
文献类型：
--
作者：
N. G. Trillos;D. Sanz
我们考虑恢复在未知域 $M$ 上制定的微分方程的函数输入的问题。我们假设可以访问离散域 $M_n=\{x_1, \dots, x_n\} \subset M$，并且可以访问这些点的 $p\le n$ 处的输出解的噪声测量。我们引入了基于图的贝叶斯逆问题，并表明 $M_n$ 中函数的图后验度量在大 $n$ 限制下收敛到 $M$ 中函数的后验，该后验函数解决了已知域的贝叶斯逆问题。
证明依赖于贝叶斯更新的变分公式，以及用于研究点云上函数测度收敛于连续统上函数测度的新拓扑。我们的框架、技术和结果可以为机器学习中基于图的任务的鲁棒不确定性量化奠定基础。这些想法是在恢复未知流形上热方程初始条件的具体设置中提出的。
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.
影响因子：
5.7
作者：
Agapiou, S.;Papaspiliopoulos, O.;Stuart, A. M.
通讯作者：
Stuart, A. M.