Abstract:This article is concerned with sampling from Gibbs distributions $\pi(x)\propto e^{-U(x)}$ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials $U(x)$ which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density $\pi$ in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate $\pi$ for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.
| Subjects: | Numerical Analysis (math.NA); Optimization and Control (math.OC) |
| MSC classes: | 60J20, 68U10, 94A08 |
| ACM classes: | G.3; I.4.5 |
| Cite as: | arXiv:2411.12051 [math.NA] |
| (or arXiv:2411.12051v2 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2411.12051 arXiv-issued DOI via DataCite |
From: Andreas Habring [view email]
[v1]
Mon, 18 Nov 2024 20:44:18 UTC (1,828 KB)
[v2]
Fri, 22 May 2026 13:26:10 UTC (6,782 KB)
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