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.

我们介绍了一种新的平均场常微分方程和相应的相互作用粒子系统，用于从非标准化目标密度或贝叶斯后验中进行采样。这些相互作用粒子系统是无梯度的，具有封闭形式，并且仅需要能够从参考密度中进行采样并计算（非标准化的）目标与参考密度的比值。通过求解输运样本沿两个密度的几何混合的速度场的泊松方程获得平均场常微分方程，这是特定的Fisher-Rao梯度流路径。我们使用再生核希尔伯特空间拟设来表示速度场，使得泊松方程易于处理，并使我们能够将得到的平均场常微分方程离散化成有限样本，形成一个简单的相互作用粒子系统。平均场常微分方程还可以从离散时间角度推导出来，作为Monge-Ampere方程在一个被称为样本驱动最优输运的框架中连续线性化的极限。我们在实证中证明，我们的相互作用粒子系统能够从具有不同特性的分布中生成高质量的样本。

基于核费舍尔流的单位时间采样