论文标题
Riemannian Langevin算法的收敛性
Convergence of the Riemannian Langevin Algorithm
论文作者
论文摘要
我们研究了Riemannian Langevin算法，这些问题是从具有公制$ g $的歧管上的自然措施的分布中进行采样的问题。我们假设目标密度满足了对Log-Sobolev的不平等，并证明了未调整的Langevin算法的流形概括迅速收敛到Hessian歧管的$ν$。这使我们能够减少$ {\ bf r}^n $在$ {\ bf r}^n $中采样的问题，以对适当的歧管进行平滑密度，同时仅需要访问对数密度的梯度，而这反过来又需要从自然的布朗尼运动中采样。我们的主要分析工具是（1）自我符合到流形的扩展，以及（2）在歧管上界限平滑度的随机方法。我们方法的一种特殊情况是通过使用对数屏障定义的度量来对限制多面体的等化密度进行采样。
We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density $ν$ with respect to the natural measure on a manifold with metric $g$. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to $ν$ for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in ${\bf R}^n$ to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of the log-density, and this, in turn, to sampling from the natural Brownian motion on the manifold. Our main analytic tools are (1) an extension of self-concordance to manifolds, and (2) a stochastic approach to bounding smoothness on manifolds. A special case of our approach is sampling isoperimetric densities restricted to polytopes by using the metric defined by the logarithmic barrier.
