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We introduce a new mean-field ODE and corresponding interacting particlesystems 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 computethe (unnormalized) target-to-reference density ratio. The mean-field ODE isobtained by solving a Poisson equation for a velocity field that transportssamples along the geometric mixture of the two densities, which is the path ofa particular Fisher-Rao gradient flow. We employ a reproducing kernel Hilbertspace ansatz for the velocity field, which makes the Poisson equation tractableand enables us to discretize the resulting mean-field ODE over finite samples,as a simple interacting particle system. The mean-field ODE can be additionallybe derived from a discrete-time perspective as the limit of successivelinearizations of the Monge-Amp\`ere equations within a framework known assample-driven optimal transport. We demonstrate empirically that ourinteracting particle systems can produce high-quality samples fromdistributions with varying characteristics.

Sampling in Unit Time with Kernel Fisher-Rao Flow