Interaction-Force Transport Gradient Flows
<|reference_start|>Interaction-Force Transport Gradient Flows: 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.<|reference_end|>
@article{gladin2024interaction-force,
    title={Interaction-Force Transport Gradient Flows},
    author={Egor Gladin, Pavel Dvurechensky, Alexander Mielke, Jia-Jie Zhu},
    journal={arXiv preprint arXiv:2405.17075},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.17075},
    primaryClass={cs.LG math.AP stat.ML}
}