Operator normed adaptive Laplace mechanism: A manifold-based approach for differential privacy

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Abstract

Modern computational intelligence and data science applications frequently operate on complex, non-Euclidean domains. Applying differential privacy (DP) to these structures requires mechanisms that respect their underlying geometry. Instance-adaptive mechanisms in DP improve utility over global worst-case bounds but often require heavy-tailed noise, leading to numerical instability and unbounded variance. We introduced a geometric framework that lifted sensitivity analysis to the tangent bundle of a Riemannian manifold. The proposed directional local sensitivity operator captured maximal directional derivatives along minimizing geodesics, enabling our operator normed adaptive Laplace mechanism (ON–ALM). By enforcing a β-log-Lipschitz stability condition on the regularized sensitivity, noise was precisely calibrated via directional operator norms. Under (ε, δ)-DP, extracting a typical set allowed the use of light-tailed Laplace noise, achieving strictly bounded variance, improved numerical stability, and high utility for geometric and directional queries.

Keywords

differential privacy / geometric sensitivity / operator norm / manifold-based modelling / Laplace mechanisms / computational intelligence / next-generation data science

CHANGE HISTORY

A Correction to this paper has been published:
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References

[1] C. Dwork, F. McSherry, K. Nissim, A. Smith, Calibrating noise to sensitivity in private data analysis, in Theory of Cryptography Conference (TCC) , Springer, Berlin, Heidelberg, (2006), 265-284.

[2] K. Nissim, S. Raskhodnikova, A. Smith, Smooth sensitivity and sampling in private data analysis, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, (2007), 75-84.

[3] D. Durfee, Unifying Laplace mechanism with instance optimality in differential privacy, preprint, arXiv:2505.02798.

[4] J. Dong, A. Roth, W. J. Su, Gaussian differential privacy, J. R. Stat. Soc. B, 84 (2022), 3-37. https://doi.org/10.1111/rssb.12454

[5] M. Reimherr, J. Awan, KNG: The K-norm gradient mechanism, in Advances in Neural Information Processing Systems, 32 (2019).

[6] A. Han, B. Mishra, P. Jawanpuria, J. Gao, Differentially private Riemannian optimization, Mach. Learn., 113 (2024), 1133-1161. https://doi.org/10.1007/s10994-023-06508-5

[7] C. Li, S. Long, H. Liu, Y. Choi, H. Sekiya, Z. Li, Enhancing sparse mobile crowdsensing with manifold optimization and differential privacy, IEEE Trans. Inf. Forensics Secur., 19 (2024), 6070-6083. https://doi.org/10.1109/TIFS.2024.3407668

[8] Z. Huang, W. Huang, P. Jawanpuria, B. Mishra, Federated learning on Riemannian manifolds with differential privacy, preprint, arXiv:2404.10029.

[9] S. Garg, V. Torra, Privacy in manifolds: Combining k-anonymity with differential privacy on Fréchet means, Comput. Secur., 144 (2024), 103983. https://doi.org/10.1016/j.cose.2024.103983

[10] P. He, L. Tang, M. A. Rahimian, J. Joshi, Conformal-DP: Differential privacy on Riemannian manifolds via conformal transformation, preprint, arXiv:2504.20941.

[11] M. P. do Carmo, Riemannian Geometry , Birkhäuser, Boston, 1992.

[12] J. M. Lee, Introduction to Riemannian Manifolds , 2nd edition, Springer, Cham, 2018.

[13] G. Casella, R. L. Berger, Statistical Inference , 2nd edition, Thomson Learning, Belmont, CA, 2002.

[14] S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Struct. Algorithms, 22 (2003), 60-65. https://doi.org/10.1002/rsa.10073

[15] C. Dwork, G. N. Rothblum, S. Vadhan, Boosting and differential privacy, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, (2010), 51-60.