Efficient Sampling on Riemannian Manifolds via Langevin MCMC
Authors: Xiang Cheng, Jingzhao Zhang, Suvrit Sra
NeurIPS 2022 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | We study the task of efficiently sampling from a Gibbs distribution dπ = e hdvolg over a Riemannian manifold M via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming h is Lipschitz and M has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within ε-Wasserstein distance of π after O(ε 2) steps, which matches the iteration complexity for Euclidean Langevin MCMC. |
| Researcher Affiliation | Academia | Xiang Cheng Massachusetts Institute of Technology x.cheng@berkeley.edu Jingzhao Zhang Tsinghua University jzhzhang@mit.edu Suvrit Sra Massachusetts Institute of Technology suvrit@mit.edu |
| Pseudocode | No | The paper describes mathematical models and processes but does not include any pseudocode or algorithm blocks. |
| Open Source Code | No | The paper does not provide any explicit statement about open-source code release or a link to a code repository. |
| Open Datasets | No | This is a theoretical paper and does not involve experimental evaluation on datasets. Therefore, no information about publicly available datasets is provided. |
| Dataset Splits | No | This is a theoretical paper and does not involve experimental evaluation on datasets, so there are no training/validation/test splits mentioned. |
| Hardware Specification | No | The paper is theoretical and does not describe computational experiments, so no hardware specifications are mentioned. |
| Software Dependencies | No | The paper is theoretical and does not specify software dependencies with version numbers for experimental reproducibility. |
| Experiment Setup | No | The paper is theoretical and does not describe an experimental setup, hyperparameters, or system-level training settings. |