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

Zihao Ren, Lei Wang, Deming Yuan, Guodong Shi

TL;DR

This paper addresses privacy protection for data constrained to affine manifolds in linear dynamical systems. It introduces a manifold-aware DP framework, derives necessary and sufficient conditions for Gaussian and Laplace noise mechanisms with structured (non–iid) noise, and provides two concrete design algorithms to meet a prescribed privacy budget. The results are demonstrated through applications to cloud-based control and average consensus, including numerical validations that reveal the privacy–utility trade-offs and the advantages of geometrically structured noise. The framework offers a unified approach to achieving differential privacy under intrinsic geometric constraints, with potential extensions to non-linear settings and alternative privacy metrics.

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.

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

TL;DR

This paper addresses privacy protection for data constrained to affine manifolds in linear dynamical systems. It introduces a manifold-aware DP framework, derives necessary and sufficient conditions for Gaussian and Laplace noise mechanisms with structured (non–iid) noise, and provides two concrete design algorithms to meet a prescribed privacy budget. The results are demonstrated through applications to cloud-based control and average consensus, including numerical validations that reveal the privacy–utility trade-offs and the advantages of geometrically structured noise. The framework offers a unified approach to achieving differential privacy under intrinsic geometric constraints, with potential extensions to non-linear settings and alternative privacy metrics.

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.
Paper Structure (29 sections, 6 theorems, 72 equations, 7 figures, 2 algorithms)

This paper contains 29 sections, 6 theorems, 72 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Differential Privacy
  4. Mechanisms over Affine Manifolds: A Motivating Example
  5. Differential Privacy over Affine Manifolds
  6. Differential Privacy Conditions on Manifolds
  7. Gaussian Mechanism
  8. Laplace Mechanism
  9. Geometrically Structured Mechanism Design
  10. Geometrically Structured Gaussian Mechanism Design
  11. Structured Laplace Mechanism Design
  12. Differentially Private Control Systems
  13. Cloud-based Control Systems
  14. Differentially Private Cloud-based Control
  15. Numerical Validations
  16. ...and 14 more sections

Key Result

Theorem 1

The mechanism $\mathscr{M}$ in (eq:mech-M) with $\bm{\eta}\sim\mathcal{N}(0,1)^r$ achieves $(\epsilon,\delta)$-differential privacy over $\mathcal{C}_d$ under $\mu$-adjacency if and only if there hold

Figures (7)

  • Figure 1: Differentially private cloud-based control scheme (e.g., tanaka2017directed).
  • Figure 2: Tracking errors $p(t)-p_r(t)$ and $v(t)-v_r(t)$.
  • Figure 3: Mean-square tracking errors $p(t)-p_r(t)$ and $v(t)-v_r(t)$ under different privacy requirements $\epsilon$ and structured noises ($r=1$ for the structured noises following Algorithm 1 and $r=T$ for i.i.d. noises).
  • Figure 4: Communication graph $\rm G$.
  • Figure 5: Trajectories of $\mathbb{E}[x_i(t)-x^{\star}]$ with Gaussian noises (500 samples).
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 2 more