On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy

We consider the problem of sampling from an unnormalized Boltzmann/ Gibbs density, π(θ) exp V(θ),θ Θ Rd, where the normalization constant is unknown (and/or intractable) and only the potential function V (and typically its derivatives) can be evaluated. This problem arises across various domains in Bayesian inference, statistical physics, and modern machine learning. A common variational perspective on sampling is to characterize the target distribution π as the unique minimizer of a functional (typically a divergence functional) over the space of probability measures. From this viewpoint, sampling can be formulated as evolving an initial distribution ρ0 toward π via the gradient flow of this functional under a suitable geometric structure on the space of probability measures. In this paper, we focus on a gradient flow based sampling methodology built from the spherical Hellinger Kantorovich (SHK), also known as the Wasserstein Fisher Rao (WFR), geometry on the space of probability measures (Kondratyev and Vorotnikov, 2019; Liero et al., 2018; Chizat et al., 2015). When the variational objective is the exclusive KL divergence ρ 7 KL(ρ π), the SHK gradient flow generates a time-indexed family of marginals {ρt}t 0 (initialized at ρ0 P2(Θ)) that evolves according to the continuity reaction equation (4). This evolution is equivalent to the birth-death Langevin dynamics introduced in Lu et al. (2019) .