Interaction-Force Transport Gradient Flows
CoRR(2024)
Abstract
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.
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