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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2604.20761 |
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| _version_ | 1866918462174003200 |
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| author | Chang, Xiaotian Jiang, Yangdi Mostajeran, Cyrus Hu, Qirui |
| author_facet | Chang, Xiaotian Jiang, Yangdi Mostajeran, Cyrus Hu, Qirui |
| contents | In this paper, we develop a novel privacy mechanism for Riemannian manifold-valued data. Our key contribution lies in uncovering unexpected connections among geometric analysis, heat diffusion models, and differential privacy (DP). We characterize the Renyi divergence via dimension-free Harnack inequalities on Riemannian manifolds and establish Renyi differential privacy guarantees governed by Ricci curvature. For manifolds with nonnegative Ricci curvature, we propose a mechanism based on heat diffusion. In contrast, for general manifolds we introduce a Langevin-process-based approach that yields intrinsic mechanisms supporting normalization-free sampling and continuous privacy-utility trade-offs. We derive detailed utility analyses for both mechanisms. As a statistical application, we develop privacy-preserving estimation of the generalized Frechet mean, including nontrivial sensitivity analysis and phase transition characterizations. Numerical experiments further demonstrate the advantages of the proposed DP mechanisms over existing approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20761 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms Chang, Xiaotian Jiang, Yangdi Mostajeran, Cyrus Hu, Qirui Machine Learning Methodology In this paper, we develop a novel privacy mechanism for Riemannian manifold-valued data. Our key contribution lies in uncovering unexpected connections among geometric analysis, heat diffusion models, and differential privacy (DP). We characterize the Renyi divergence via dimension-free Harnack inequalities on Riemannian manifolds and establish Renyi differential privacy guarantees governed by Ricci curvature. For manifolds with nonnegative Ricci curvature, we propose a mechanism based on heat diffusion. In contrast, for general manifolds we introduce a Langevin-process-based approach that yields intrinsic mechanisms supporting normalization-free sampling and continuous privacy-utility trade-offs. We derive detailed utility analyses for both mechanisms. As a statistical application, we develop privacy-preserving estimation of the generalized Frechet mean, including nontrivial sensitivity analysis and phase transition characterizations. Numerical experiments further demonstrate the advantages of the proposed DP mechanisms over existing approaches. |
| title | Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms |
| topic | Machine Learning Methodology |
| url | https://arxiv.org/abs/2604.20761 |