Spherical Hellinger-Kantorovich geometry

Perturbation Theory for Spherical Hellinger-Kantorovich Flows with Differential Privacy Guarantees

This paper develops a perturbation theory for Spherical Hellinger-Kantorovich (SHK) gradient flows, which couple transport and reaction dynamics and coincide with birth-death Langevin dynamics. The authors derive dimension-free bounds on log-likelihood ratios and Rényi/KL divergences when two potentials differ, quantifying how perturbations propagate over time. These results are applied to differential privacy: the likelihood-ratio control yields explicit Pure-DP guarantees for SHK-based samplers implementing the exponential mechanism, while KL bounds provide Approximate-DP certificates. A utility bound is also derived that separates intrinsic exponential-mechanism suboptimality from finite-time sampling error.