AI 中文总结
研究给定概率密度\(\mu\)时，能否选强凸函数\(u\)使镜像朗之万扩散指数收敛到平衡态，基于李雅普诺夫函数方法给出条件。还引入马尔可夫链逼近，证明其在\(\chi^2\)中有与扩散时间尺度一致的收敛速率。
AI 中文摘要
给定一个强凸函数\(u\)，在\(R^d\)上配备由黑塞矩阵\(\nabla^2 u\)给出的黎曼度量，即所谓的黑塞流形。给定概率密度\(\mu\)，可运行该流形内在的具有平稳分布\(\mu\)的朗之万扩散，此类（黑塞）流形值朗之万扩散称为镜像朗之万扩散（MLD），近来颇受关注。我们探讨的问题之一是，给定\(\mu\)，能否选择\(u\)使MLD指数收敛到平衡态，特别是当\(\mu\)并非强对数凹时。我们的结果基于李雅普诺夫函数方法，给出了MLD满足庞加莱不等式或对数索伯列夫不等式的充分条件，进而意味着指数收敛。我们还引入了由具有平稳分布\(\mu\)的两步吉布斯采样器给出的MLD的马尔可夫链逼近。该马尔可夫链是arXiv:2307.16421中引入的Sinkhorn马尔可夫链的变体，据推测它收敛到MLD的一个时间非齐次推广。在适当假设下，我们证明该马尔可夫链在\(\chi^2\)中有保证的收敛速率，且与扩散时间尺度一致。我们的证明基于熵最优传输和强数据处理不等式的思想。
英文摘要
Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $μ$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $μ$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $μ$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $μ$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincaré or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $μ$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $χ^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.
Comments37 pages
