Title: Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows Show Chinese title

Title: Wasserstein-Fisher-Rao梯度流的顺序蒙特卡罗近似

Abstract: We consider the problem of sampling from a probability distribution $\pi$. It is well known that this can be written as an optimisation problem over the space of probability distribution in which we aim to minimise the Kullback--Leibler divergence from $\pi$. We consider several partial differential equations (PDEs) whose solution is a minimiser of the Kullback--Leibler divergence from $\pi$ and connect them to well-known Monte Carlo algorithms. We focus in particular on PDEs obtained by considering the Wasserstein--Fisher--Rao geometry over the space of probabilities and show that these lead to a natural implementation using importance sampling and sequential Monte Carlo. We propose a novel algorithm to approximate the Wasserstein--Fisher--Rao flow of the Kullback--Leibler divergence which empirically outperforms the current state-of-the-art. We study tempered versions of these PDEs obtained by replacing the target distribution with a geometric mixture of initial and target distribution and show that these do not lead to a convergence speed up. Show Chinese abstract

Abstract: 我们研究了从概率分布$\pi$抽样的问题。众所周知，这可以写成概率空间上的一个优化问题，其中我们旨在最小化 Kullback-Leibler 散度与$\pi$的距离。我们考虑了几种偏微分方程（PDEs），其解是最小化 Kullback-Leibler 散度与$\pi$距离的极小值点，并将它们与著名的蒙特卡罗算法联系起来。我们特别关注通过在概率空间上考虑 Wasserstein-Fisher-Rao 几何得到的 PDEs，并表明这些可以自然地通过重要性采样和序列蒙特卡罗方法实现。我们提出了一种新的算法来近似 Kullback-Leibler 散度的 Wasserstein-Fisher-Rao 流，该算法在实验中优于当前最先进的方法。我们研究了这些 PDEs 的温化版本，通过用初始分布和目标分布的几何混合替换目标分布得到，并证明这些并不会加速收敛速度。