This paper presents a new type of gradient flow geometries over non-negative and probability measures motivated via a principled construction that combines the optimal transport and interaction forces modeled by reproducing kernels. Concretely, we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD Riemannian metric tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization studied by Arbel et al. [2019]. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy.

翻译：本文提出了一种新型的非负测度与概率测度上的梯度流几何结构，其构建原理融合了最优传输与由再生核建模的相互作用力。具体而言，我们通过Wasserstein度量张量与球面最大均值差异黎曼度量张量的下确界卷积，提出了相互作用力传输梯度流及其球面变体。随后，我们基于质量守恒的球面相互作用力传输梯度流的JKO分裂格式，发展了一种基于粒子的优化算法。最后，针对Arbel等人[2019]所研究的最大均值差异最小化采样任务，我们应用相互作用力传输梯度流，不仅提供了理论上的全局指数收敛性保证，还给出了实证模拟结果。此外，我们证明球面相互作用力传输梯度流兼具两者优势，为最大均值差异能量与KL能量均提供了全局指数收敛性保证。