Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities

The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.

💡 Research Summary

The paper investigates the dynamics of probability density functions when the underlying geometry is given by the Fisher‑Rao metric, in contrast to the more widely studied Wasserstein setting. The authors begin by recalling that many sampling problems can be cast as the evolution of a density ρ(t) toward a target density ρ*∝exp(−V), and that the Langevin diffusion and its associated Fokker‑Planck equation are precisely the Wasserstein gradient flow of the Kullback–Leibler (KL) divergence. In the Wasserstein framework, exponential convergence of the flow follows from a log‑Sobolev inequality, which is equivalent to a gradient‑dominance (Polyak‑Łojasiewicz) condition. However, the convergence rate depends on the log‑Sobolev constant of ρ*, which can be arbitrarily small for anisotropic or multimodal targets.

The Fisher‑Rao metric is introduced as g_FR_ρ(σ₁,σ₂)=∫σ₁σ₂ρ dθ, the spherical restriction of the Hellinger distance. Under this metric, the KL divergence generates a non‑local ordinary differential equation for each point θ: ∂ₜρₜ = −ρₜ