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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 F\u00f6rstner 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.<\/jats:p>","DOI":"10.1137\/140977308","type":"journal-article","created":{"date-parts":[[2015,11,3]],"date-time":"2015-11-03T14:07:59Z","timestamp":1446559679000},"page":"A2451-A2487","source":"Crossref","is-referenced-by-count":94,"title":["Optimal Low-rank Approximations of Bayesian Linear Inverse Problems"],"prefix":"10.1137","volume":"37","author":[{"given":"Alessio","family":"Spantini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Antti","family":"Solonen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tiangang","family":"Cui","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"James","family":"Martin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis","family":"Tenorio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Youssef","family":"Marzouk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,11,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718133.ch16"},{"key":"atypb2","first-page":"217","volume":"35","author":"Auvinen H.","year":"2009","journal-title":"Electron. 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