Journal Article

PUBDB-2024-05071

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Proximal Langevin Sampling with Inexact Proximal Mapping

Ehrhardt, M. J. ; Kuger, L. (Corresponding author)Extern*DESY* ; Schönlieb, C.-B.

2024
SIAM Philadelphia, Pa.

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Please use a persistent id in citations: doi:10.1137/23M1593565

Abstract: In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution.For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool. If the potential defining the distribution is non-smooth, these discretizations are usually of an implicit form leading to Langevin sampling algorithms that require the evaluation of proximal operators. For some of the potentials relevant in imaging problems this is only possible approximately using an iterative scheme. We investigate the behaviour of a proximal Langevin algorithm under the presence of errors in the evaluation of proximal mappings. We generalize existing non-asymptotic and asymptotic convergence results of the exact algorithm to our inexact setting and quantify the bias between the target and the algorithm's stationary distribution due to the errors. We show that the additional bias stays bounded for bounded errors and converges to zero for decaying errors in a strongly convex setting. We apply the inexact algorithm to sample numerically from the posterior of typical imaging inverse problems in which we can only approximate the proximal operator by an iterative scheme and validate our theoretical convergence results.

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Contributing Institute(s):

1. Computational Imaging (FS-CI)

Research Program(s):

1. 623 - Data Management and Analysis (POF4-623) (POF4-623)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2024

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Database coverage:
; Clarivate Analytics Master Journal List ; Current Contents - Engineering, Computing and Technology ; Essential Science Indicators ; IF < 5 ; JCR ; SCOPUS ; Science Citation Index Expanded ; Web of Science Core Collection

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