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

Statistics > Machine Learning [Submitted on 22 May 2026] On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy Aratrika Mustafi, Soumya Mukherjee Gradient-flow sampling interprets a Gibbs distribution as the minimizer of an energy functional over probability measures and generates dynamics converging to this target. Under spherical Hellinger-Kantorovich (SHK) geometry, the flow couples transport and reaction and coincides with birth-death Langevin dynamics. In this work, we develop a perturbation theory for SHK gradient flows. For two potentials V and V^{\prime}, we compare the associated flows from a common initialization and quantify how potential discrepancies propagate over time. A uniform perturbation bound yields dimension-free, pointwise control of the log-likelihood ratio and Rényi divergence, while additional structure allows us to derive bounds for the KL divergence as well. We apply these results to approximate sampling for the exponential mechanism in differential privacy. The likelihood-ratio control provides explicit time-dependent Pure-DP guarantees for SHK-based samplers, while the KL bound yields Approximate-DP certificates via hockey-stick divergence. We also derive a utility bound separating intrinsic exponential-mechanism suboptimality from finite-time sampling error. Subjects: Machine Learning (stat.ML); Cryptography and Security (cs.CR); Machine Learning (cs.LG); Statistics Theory (math.ST) Cite as: arXiv:2605.23879 [stat.ML]   (or arXiv:2605.23879v1 [stat.ML] for this version)   https://doi.org/10.48550/arXiv.2605.23879 Focus to learn more Submission history From: Aratrika Mustafi [view email] [v1] Fri, 22 May 2026 17:38:20 UTC (1,412 KB) Access Paper: HTML (experimental) view license Current browse context: stat.ML < prev   |   next > new | recent | 2026-05 Change to browse by: cs cs.CR cs.LG math math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)