针对重尾目标的退火弹跳粒子采样器的次椭圆性
Hypocoercivity of Tempered Bouncy Particle Samplers for Heavy-Tailed Targets
AI总结：
该研究提出退火弹跳粒子采样器，利用次椭圆性框架为非对数凹、非径向对称的重尾目标建立指数收敛界，推导混合时间对维度呈多项式依赖的界，为相关采样问题提供了理论支撑。
AI中文摘要：
我们提出了退火弹跳粒子采样器（Tempered Bouncy Particle Sampler），它是弹跳粒子采样器（Bouncy Particle Sampler）的一种依赖状态的退火推广。通过适配Dolbeault、Mouhot和Schmeiser（2010）的次椭圆性（hypocoercivity）框架，我们为一大类无需对数凹或径向对称的重尾目标建立了显式非渐近指数收敛界，包括具有多项式尾部的分布。收敛率与具有相同不变分布的退火朗之万扩散的谱间隙相关，加权庞加莱不等式是关键假设。我们给出了多项式尾部和拉伸指数尾部的可接受退火选择，并推导了混合时间对维度呈多项式依赖的界。
英文摘要：
We introduce the Tempered Bouncy Particle Sampler, a state-dependently tempered generalisation of the Bouncy Particle Sampler. Adapting the hypocoercivity framework of Dolbeault, Mouhot and Schmeiser (2010), we establish explicit, non-asymptotic exponential convergence bounds for a broad class of heavy-tailed targets that need not be log-concave or radially symmetric, including distributions with polynomial tails. The convergence rate is linked to the spectral gap of a tempered Langevin diffusion with the same invariant distribution, making a weighted Poincaré inequality the key assumption. We give admissible tempering choices for polynomial and stretched-exponential tails and derive mixing-time bounds with polynomial dependence on the dimension.
