@inproceedings{
gladin2024interactionforce,
title={Interaction-Force Transport Gradient Flows},
author={Egor Gladin and Pavel Dvurechensky and Alexander Mielke and Jia-Jie Zhu},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=rPgc5brxmT}
}
This paper presents a new gradient flow dissipation geometry over non-negative and probability measures.
This is motivated by a principled construction that combines the unbalanced optimal transport and interaction forces modeled by reproducing kernels. Using a precise connection between the Hellinger geometry and the maximum mean discrepancy (MMD), we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD 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 improved empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization. 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.
Interaction-Force Transport Gradient Flows
Egor Gladin Pavel Dvurechensky Alexander Mielke Jia-Jie Zhu