# Interaction-Force Transport Gradient Flows

arXiv: https://arxiv.org/abs/2405.17075 · HTML: https://mlsanity.com/paper/2405.17075/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 58 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 23 of 23)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2301.01766/) (2023)
- [Provable Bayesian Inference via Particle Mirror Descent](https://mlsanity.com/paper/1506.03101/) (2015)
- [Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering](https://mlsanity.com/paper/1501.02056/) (2015)
- [Estimating Barycenters of Measures in High Dimensions](https://mlsanity.com/paper/2007.07105/) (2020)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [MMD-Regularized Unbalanced Optimal Transport](https://mlsanity.com/paper/2011.05001/) (2020)
- [A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling](https://mlsanity.com/paper/2202.12927/) (2022)

## Built on by (most cited first, 9 of 9)

- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals](https://mlsanity.com/paper/2501.17049/) (2025)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)
- [Kernel-based External Validity of Experimental Studies](https://mlsanity.com/paper/2610.13625/) (2026)
- [On the Optimisation Landscape of Overparameterised Weighted MMD](https://mlsanity.com/paper/2610.19861/) (2026)

## Abstract (arXiv)

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.