论文标题
通过稳定驱动的SDE近似重尾分布
Approximation of heavy-tailed distributions via stable-driven SDEs
论文作者
论文摘要
众多近似采样算法的构造是基于以下事实：某些Gibbs度量是由Brownian运动驱动的Ergodic随机微分方程（SDE）的固定分布。但是，对于某些重尾分布，可以证明相关的SDE并非指数性的千古化，并且相关的采样算法的性能可能很差。在这种情况下，最近在机器学习文献中探索的一个自然想法是，使用带有沉重的尾巴而不是布朗运动的随机过程。在本文中，我们提供了一个严格的理论框架，用于研究通过由对称（旋转不变的）$α$稳定过程驱动的Ergodic SDE近似重尾分布的问题。
Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paper we provide a rigorous theoretical framework for studying the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) $α$-stable processes.
