We introduce a new mean-field ODE and corresponding interacting particle systems for sampling from an unnormalized target density or Bayesian posterior. The interacting particle systems are gradient-free, available in closed form, and only require the ability to sample from the reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, which is the path of a particular Fisher-Rao gradient flow. We employ a reproducing kernel Hilbert space ansatz for the velocity field, which makes the Poisson equation tractable and enables us to discretize the resulting mean-field ODE over finite samples, as a simple interacting particle system. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Amp\`ere equations within a framework known as sample-driven optimal transport. We demonstrate empirically that our interacting particle systems can produce high-quality samples from distributions with varying characteristics.

翻译：我们引入一个新的平均场常微分方程和相应的交互粒子系统，用于从非归一化目标密度或贝叶斯后验中进行采样。这些交互粒子系统无需梯度、具有闭合形式解，且仅需具备从参考密度中采样以及计算（非归一化）目标-参考密度比的能力。该平均场常微分方程通过求解速度场的泊松方程获得，该速度场沿两个密度的几何混合（即特定Fisher-Rao梯度流的路径）传输样本。我们对速度场采用再生核希尔伯特空间假设，这使得泊松方程易于处理，并能将所得平均场常微分方程离散为有限样本上的简单交互粒子系统。该平均场常微分方程还可从离散时间视角推导，它是样本驱动最优传输框架下Monge-Ampère方程逐次线性化的极限。实验表明，我们的交互粒子系统能从具有不同特征的分布中生成高质量样本。