TY  - JOUR
AU  - Ehrhardt, Matthias J.
AU  - Kuger, Lorenz
AU  - Schönlieb, Carola-Bibiane
TI  - Proximal Langevin Sampling with Inexact Proximal Mapping
JO  - SIAM journal on imaging sciences
VL  - 17
IS  - 3
SN  - 1936-4954
CY  - Philadelphia, Pa.
PB  - SIAM
M1  - PUBDB-2024-05071
SP  - 1729-1760
PY  - 2024
AB  - 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.
LB  - PUB:(DE-HGF)16
UR  - <Go to ISI:>//WOS:001343420300001
DO  - DOI:10.1137/23M1593565
UR  - https://bib-pubdb1.desy.de/record/611791
ER  -