Subgradient Langevin Methods for Sampling from Non-smooth Potentials
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929357691289600 |
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| author | Habring, Andreas Holler, Martin Pock, Thomas |
| author_facet | Habring, Andreas Holler, Martin Pock, Thomas |
| contents | This paper is concerned with sampling from probability distributions $π$ on $\mathbb{R}^d$ admitting a density of the form $π(x) \propto e^{-U(x)}$, where $U(x)=F(x)+G(Kx)$ with $K$ being a linear operator and $G$ being non-differentiable. Two different methods are proposed, both employing a subgradient step with respect to $G\circ K$, but, depending on the regularity of $F$, either an explicit or an implicit gradient step with respect to $F$ can be implemented. For both methods, non-asymptotic convergence proofs are provided, with improved convergence results for more regular $F$. Further, numerical experiments are conducted for simple 2D examples, illustrating the convergence rates, and for examples of Bayesian imaging, showing the practical feasibility of the proposed methods for high dimensional data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01417 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Subgradient Langevin Methods for Sampling from Non-smooth Potentials Habring, Andreas Holler, Martin Pock, Thomas Optimization and Control Computation 65C40, 65C05, 68U10, 65C60 G.3; G.1.6 This paper is concerned with sampling from probability distributions $π$ on $\mathbb{R}^d$ admitting a density of the form $π(x) \propto e^{-U(x)}$, where $U(x)=F(x)+G(Kx)$ with $K$ being a linear operator and $G$ being non-differentiable. Two different methods are proposed, both employing a subgradient step with respect to $G\circ K$, but, depending on the regularity of $F$, either an explicit or an implicit gradient step with respect to $F$ can be implemented. For both methods, non-asymptotic convergence proofs are provided, with improved convergence results for more regular $F$. Further, numerical experiments are conducted for simple 2D examples, illustrating the convergence rates, and for examples of Bayesian imaging, showing the practical feasibility of the proposed methods for high dimensional data. |
| title | Subgradient Langevin Methods for Sampling from Non-smooth Potentials |
| topic | Optimization and Control Computation 65C40, 65C05, 68U10, 65C60 G.3; G.1.6 |
| url | https://arxiv.org/abs/2308.01417 |