Optimal low-rank approximations of Bayesian linear inverse problems

Alessio Spantini, Antti Solonen, Tiangang Cui, James Martin, Luis Tenorio, Youssef Marzouk

Research output: Contribution to journalArticleResearchpeer-review

39 Citations (Scopus)

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 loglikelihood 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 squarederror 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 highdimensional 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.

Original languageEnglish
Pages (from-to)A2451-A2487
Number of pages37
JournalSIAM Journal on Scientific Computing
Volume37
Issue number6
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Bayes risk
  • Bayesian inference
  • Covariance approximation
  • Förstner-moonen metric
  • Inverse problems
  • Low-rank approximation
  • Optimality
  • Posterior mean approximation

Cite this