Optimal Low-rank Approximations of Bayesian Linear Inverse Problems
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Spantini-2015-Optimal Low-rank.pdf
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Author(s)
Spantini, Alessio
Solonen, Antti
Cui, Tiangang
Martin, James
Tenorio, Luis
Marzouk, Youssef M.
Date Issued
November 2015
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Spantini, Alessio, Antti Solonen, Tiangang Cui, James Martin, Luis Tenorio, and Youssef Marzouk. “Optimal Low-Rank Approximations of Bayesian Linear Inverse Problems.” SIAM Journal on Scientific Computing 37, no. 6 (January 2015): A2451–A2487. © 2015 Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the Förstner metric for symmetric positive definite matrices, as well as the Kullback--Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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DOI of Published Version
https://doi.org/10.1137/140977308
