We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a 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 RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. 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 introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.

翻译：我们提出了一种新的平均场常微分方程及相应的相互作用粒子系统，用于从非归一化目标密度中进行采样。该相互作用粒子系统无需梯度计算，具有闭式形式，仅需从参考密度中采样并计算（非归一化）目标与参考密度之比的能力。平均场常微分方程通过求解泊松方程获得速度场实现，该速度场沿两种密度的几何混合路径传输样本——这正是特定Fisher-Rao梯度流的轨迹。我们采用再生核希尔伯特空间假设描述速度场，使泊松方程可解，并实现对有限样本下平均场常微分方程的离散化。从离散时间视角看，该平均场常微分方程可进一步推导为Monge-Ampère方程在样本驱动最优传输框架中逐次线性化的极限。我们提出该方法的随机变体，并通过实验表明，所提相互作用粒子系统能从多样目标分布中生成高质量样本，性能优于同类无梯度粒子系统，且与基于梯度的替代方法具有竞争力。