# Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals

arXiv: https://arxiv.org/abs/2501.17049 · HTML: https://mlsanity.com/paper/2501.17049/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 56 references that are in the map.
- First posted on arXiv 28 Jan 2025.

## Builds on (most cited first, 16 of 16)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [Fast yet Simple Natural-Gradient Descent for Variational Inference in Complex Models](https://mlsanity.com/paper/1807.04489/) (2018)
- [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)
- [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)

## Built on by (most cited first, 7 of 7)

- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Verifier-Constrained Flow Expansion for Discovery Beyond the Data](https://mlsanity.com/paper/2602.15984/) (2026)
- [Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow](https://mlsanity.com/paper/2504.20400/) (2025)
- [Gradient Flow Sampler-based Distributionally Robust Optimization](https://mlsanity.com/paper/2510.25956/) (2025)
- [Wasserstein-Cram'er-Rao Theory of Unbiased Estimation](https://mlsanity.com/paper/2511.07414/) (2025)
- [A Unified Density Operator View of Flow Control and Merging](https://mlsanity.com/paper/2602.08012/) (2026)
- [Neural Sampling with Reweighted Normalizing Flows via the Wasserstein--Fisher--Rao JKO Scheme](https://mlsanity.com/paper/2610.10278/) (2026)

## Abstract (arXiv)

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or Fisher-Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals (e.g. KL, chi2) under Otto-Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures -- where the typical log-Sobolev arguments fail -- we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the (Polyak-)Lojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning.