Differential Privacy on Affine Manifolds: Geometrically Confined Privacy in Linear Dynamical Systems

Abstract

In this paper, we present a comprehensive framework for differential privacy over affine manifolds and validate its usefulness in the contexts of differentially private cloud-based control and average consensus. We consider differential privacy mechanisms for linear queries when the input data are constrained to lie on affine manifolds, a structural property that is assumed to be available as prior knowledge to adversaries. In this setting, the definition of neighborhood adjacency must be formulated with respect to the intrinsic geometry of the manifolds. We demonstrate that such affine-manifold constraints can fundamentally alter the attainable privacy levels relative to the unconstrained case. In particular, we derive necessary and sufficient conditions under which differential privacy can be realized via structured noise injection mechanisms, wherein correlated Gaussian or Laplace noise distributions, rather than i.i.d. perturbations, are calibrated to the dataset. Based on these characterizations, we develop explicit noise calibration procedures that guarantee the tight realization of any prescribed privacy budget with a matching noise magnitude. Finally, we show that the proposed framework admits direct applications to linear dynamical systems ranging from differentially private cloud-based control to privacy-preserving average consensus, all of which naturally involve affine-manifold constraints. The established theoretical results are illustrated through numerical examples.