Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals (2501.17049v1)

Abstract: 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, $\chi^2$) 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-)\L{}ojasiewicz-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.

• The paper characterizes HK gradient flows that merge transport and reaction processes, proving global exponential decay for entropy functionals.
• It employs advanced inequalities, including Polyak-Łojasiewicz, to extend dissipation estimates beyond the classical Otto-Wasserstein framework.
• The findings lay a robust theoretical foundation with implications for variational inference, optimization, and machine learning.

An Analysis of Hellinger-Kantorovich Gradient Flows for Entropy Functionals

The paper "Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals" addresses the mathematical underpinnings and implications of certain gradient flows in a metric setting that combines transport and reaction (birth-death) processes. Traditional gradient flows have largely been studied within the Otto-Wasserstein geometric framework, which focuses on transport mechanisms for probability measures. The present work extends this framework by incorporating the Hellinger (or Fisher-Rao) mechanics, emphasizing the creation and destruction of mass, to form a unified approach using the Hellinger-Kantorovich (HK) geometry.

Key Contributions

1. Characterization of Gradient Flows: The paper characterizes gradient flows over positive and probability measures under the HK geometry. This geometry effectively merges the Otto-Wasserstein transport mechanism and the Hellinger reaction mechanism, catering to unbalanced transport scenarios where mass can be both transferred and transformed.
2. Entropy Functionals: The primary focus is on the global exponential decay of entropy functionals, such as Kullback-Leibler (KL) divergence and $\chi^2$-divergences, when driven by HK gradient flows. The paper uses specialized mathematical tools such as shape-mass decomposition and \L{}ojasiewicz-type functional inequalities to achieve analytical results that are not readily accessible through traditional methods.
3. Mathematical Framework and Inequalities: The research provides a robust theoretical framework characterizing gradient flows. It leverages Polyak-\L{}ojasiewicz inequalities and extends classical dissipation estimates to new settings. The inequalities are employed to demonstrate global convergence results and establish conditions under which these entropy functionals decay exponentially in the context of HK geometry.
4. Extension to the Hellinger-Kantorovich Framework: The authors indicate the natural generalization from Otto-Wasserstein to HK geometry by providing results that underline its advantageous properties. While standard class results rely on conditions like the Bakry-Émery criterion for functional inequalities in transport-only scenarios, this paper manages to present conditions for convergence and decay in more complex balanced transport scenarios.
5. Numerical and Theoretical Implications: The results underline numerous significant implications for statistical inference and optimization algorithms, particularly those involved in variational inference, machine learning, and sampling. The HK framework's applicability could redefine efficient algorithmic strategies for varied domains in computational statistics.

Future Developments

The synthesis of transport and reaction dynamics in gradient flows highlighted by this paper opens several avenues for future research. Potential directions include the exploration of more complex settings of entropy functional decays, the impact of these dynamics on real-world data sets, and the practical implementation of these theoretical findings in high-dimensional computational algorithms. Additionally, further examination of specific structures within the HK framework that promote even stronger convergence results could directly influence areas like variational inference and improve existing approaches to divergence minimization problems in machine learning.

The advancement presented in this paper reveals an intricate layer of mathematical theory that enriches the understanding of entropy decay in a more generalized metric space. Bridging the gap between pure transport mechanisms and reactive modifications, the Hellinger-Kantorovich gradient flows provide a compelling foundation for both theoretical exploration and practical application.