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

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 $ \beta $-log-Lipschitz stability condition on the regularized sensitivity, noise was precisely calibrated via directional operator norms. Under $ (\varepsilon, \delta) $-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.