提出新方法实现受限空间采样，理论保证收敛性。
Functional Gradient Flows for Constrained Sampling
- 引入边界条件控制梯度流，约束粒子在指定域内
- 理论证明连续时间下总变差收敛，适用于复杂约束
- 适合需要精确边界控制的采样场景，如物理模拟
近期研究从马尔可夫链蒙特卡洛（MCMC）与变分推断（VI）的统一梯度流视角出发，提出了粒子型变分推断（ParVI）方法，兼具两者优势。典型方法如斯坦因变分梯度下降（SVGD）在再生核希尔伯特空间（RKHS）中近似梯度流，但近年尝试用神经网络等更丰富的函数空间替代RKHS。然而这些方法多针对无约束域采样。本文提出一种通用解决方案：通过为梯度流引入边界条件，将粒子限制在特定域内，从而构建新的受限采样功能梯度流方法（CFG），并证明其在总变差（TV）意义下的连续时间收敛性。同时提出新颖数值策略处理由域约束引起的边界积分项。理论与实验均验证了该框架的有效性。
原文摘要 · Abstract (English)
Recently, through a unified gradient flow perspective of Markov chain Monte Carlo (MCMC) and variational inference (VI), particle-based variational inference methods (ParVIs) have been proposed that tend to combine the best of both worlds. While typical ParVIs such as Stein Variational Gradient Descent (SVGD) approximate the gradient flow within a reproducing kernel Hilbert space (RKHS), many attempts have been made recently to replace RKHS with more expressive function spaces, such as neural networks. While successful, these methods are mainly designed for sampling from unconstrained domains. In this paper, we offer a general solution to constrained sampling by introducing a boundary condition for the gradient flow which would confine the particles within the specific domain. This allows us to propose a new functional gradient ParVI method for constrained sampling, called constrained functional gradient flow (CFG), with provable continuous-time convergence in total variation (TV). We also present novel numerical strategies to handle the boundary integral term arising from the domain constraints. Our theory and experiments demonstrate the effectiveness of the proposed framework.
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