Gradient flows of the Kullback--Leibler (KL) divergence, such as the Fokker--Planck equation and Stein Variational Gradient Descent, evolve a distribution toward a target density known only up to a normalizing constant. We introduce new gradient flows of the KL divergence with a remarkable combination of properties: they admit accurate interacting-particle approximations in high dimensions, and the per-step cost scales linearly in both the number of particles and the dimension. These gradient flows are based on new transportation-based Riemannian geometries on the space of probability measures: the Radon--Wasserstein geometry and the related Regularized Radon--Wasserstein (RRW) geometry. We define these geometries using the Radon transform so that the gradient-flow velocities depend only on one-dimensional projections. This yields interacting-particle-based algorithms whose per-step cost follows from efficient Fast Fourier Transform-based evaluation of the required 1D convolutions. We additionally provide numerical experiments that study the performance of the proposed algorithms and compare convergence behavior and quantization. Finally, we prove some theoretical results including well-posedness of the flows and long-time convergence guarantees for the RRW flow.

翻译：Kullback--Leibler (KL) 散度的梯度流，例如Fokker--Planck方程和Stein变分梯度下降，可将分布演化至仅知其未归一化常数的目标密度。我们引入KL散度的一类新型梯度流，其具备一系列卓越特性：它们在高维空间中允许精确的交互粒子近似，且单步计算成本随粒子数与维度均呈线性增长。这些梯度流基于概率测度空间上新型的基于传输的黎曼几何结构：Radon--Wasserstein几何及其相关的正则化Radon--Wasserstein (RRW) 几何。我们通过Radon变换定义这些几何结构，使得梯度流速度仅依赖于一维投影。由此产生的基于交互粒子的算法，其单步计算成本得益于基于快速傅里叶变换的高效一维卷积计算。我们还通过数值实验研究了所提算法的性能，并比较了收敛行为与量化效果。最后，我们证明了若干理论结果，包括流动的适定性以及RRW流动的长时间收敛性保证。