基于带温度参数的薛定谔-弗尔默采样的多模态采样
Multimodal sampling via Schrödinger-Föllmer samplers with temperatures
AI总结：
本文通过引入温度参数并基于薛定谔-弗尔默扩散的欧拉离散化提出新采样器，在L^2-Wasserstein距离下获得O(h)的改进收敛率，数值实验表明高温对多模态分布采样至关重要且性能优于Langevin采样器。
摘要：
从复杂高维分布中生成样本在统计物理、贝叶斯推断、科学计算和机器学习等各个科学领域普遍存在。最近，Huang等人（IEEE Trans. Inform. Theory, 2025）基于在单位区间$[0, 1]$上演化的薛定谔-弗尔默扩散的欧拉离散化，提出了新的薛定谔-弗尔默采样器（SFS）。在那里，对于均匀时间步长$h>0$的欧拉离散化，在$L^2$-Wasserstein距离下获得了$\mathcal{O}(\sqrt{h})$阶的收敛率。通过引入温度参数，本文基于带温度的薛定谔-弗尔默过程的欧拉离散化提出了不同的采样器。数值实验表明，高温至关重要，特别是在从多模态分布采样时。此外，针对时间离散化开发了一种新的误差分析方法，并在漂移满足一定光滑性条件下，在$L^2$-Wasserstein距离下获得了$\mathcal{O}(h)$阶的改进收敛率。这显著改进了前述论文中现有的半阶收敛性。与Langevin采样器不同，SFS无需梯度，在单位区间$[0, 1]$上工作，且不需要任何遍历性。数值实验验证了收敛率，并表明SFS在采样多模态分布时显著优于普通Langevin采样器。
英文摘要：
Generating samples from complex and high-dimensional distributions is ubiquitous in various scientific fields of statistical physics, Bayesian inference, scientific computing and machine learning. Very recently, Huang et al. (IEEE Trans. Inform. Theory, 2025) proposed new Schrödinger-Föllmer samplers (SFS), based on the Euler discretization of the Schrödinger-Föllmer diffusion evolving on the unit interval $[0, 1]$. There, a convergence rate of order $\mathcal{O}(\sqrt{h})$ in the $L^2$-Wasserstein distance was obtained for the Euler discretization with a uniform time step-size $h>0$. By introducing a temperature parameter, different samplers are proposed in this paper, based on the Euler discretization of the Schrödinger-Föllmer process with temperatures. As revealed by numerical experiments, high temperatures are vital, particularly in sampling from multimodal distributions. Further, a novel approach of error analysis is developed for the time discretization and an enhanced convergence rate of order $\mathcal{O}(h)$ is obtained in the $L^2$-Wasserstein distance, under certain smoothness conditions on the drift. This significantly improves the existing order-half convergence in the aforementioned paper. Unlike Langevin samplers, SFS is gradient-free, works in a unit interval $[0, 1]$ and does not require any ergodicity. Numerical experiments confirm the convergence rate and show that, the SFS substantially outperforms vanilla Langevin samplers, particularly in sampling from multimodal distributions.
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