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Initial State Privacy of Nonlinear Systems on Riemannian Manifolds

Le Liu, Yu Kawano, Antai Xie, Ming Cao

TL;DR

The paper tackles initial state privacy for discrete-time nonlinear systems on Riemannian manifolds by defining an initial-state adjacency via the Riemannian distance $d_{ ilde{P}}$ and formulating differential privacy through incremental output boundedness. It develops a sufficient condition that uses time-varying Laplacian noise, with diversity $b_k$, to achieve $\epsilon_k$-differential privacy, verified via contraction analysis. The framework is extended to protect system parameters by augmenting the state with parameter variables, and demonstrated through a scalar/exo-system formulation. A numerical example demonstrates the privacy-utility trade-off and confirms the theoretical guarantees, highlighting practical impact for privacy-preserving control in geometry-constrained dynamical systems.

Abstract

In this paper, we investigate initial state privacy protection for discrete-time nonlinear closed systems. By capturing Riemannian geometric structures inherent in such privacy challenges, we refine the concept of differential privacy through the introduction of an initial state adjacency set based on Riemannian distances. A new differential privacy condition is formulated using incremental output boundedness, enabling the design of time-varying Laplacian noise to achieve specified privacy guarantees. The proposed framework extends beyond initial state protection to also cover system parameter privacy, which is demonstrated as a special application.

Initial State Privacy of Nonlinear Systems on Riemannian Manifolds

TL;DR

The paper tackles initial state privacy for discrete-time nonlinear systems on Riemannian manifolds by defining an initial-state adjacency via the Riemannian distance and formulating differential privacy through incremental output boundedness. It develops a sufficient condition that uses time-varying Laplacian noise, with diversity , to achieve -differential privacy, verified via contraction analysis. The framework is extended to protect system parameters by augmenting the state with parameter variables, and demonstrated through a scalar/exo-system formulation. A numerical example demonstrates the privacy-utility trade-off and confirms the theoretical guarantees, highlighting practical impact for privacy-preserving control in geometry-constrained dynamical systems.

Abstract

In this paper, we investigate initial state privacy protection for discrete-time nonlinear closed systems. By capturing Riemannian geometric structures inherent in such privacy challenges, we refine the concept of differential privacy through the introduction of an initial state adjacency set based on Riemannian distances. A new differential privacy condition is formulated using incremental output boundedness, enabling the design of time-varying Laplacian noise to achieve specified privacy guarantees. The proposed framework extends beyond initial state protection to also cover system parameter privacy, which is demonstrated as a special application.

Paper Structure

This paper contains 6 sections, 3 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential Privacy Analysis
  4. Application in System Parameter Protection
  5. A Numerical Example
  6. Conclusions

Key Result

Theorem 3.2

Given $\zeta > 0$, a mechanism eq:mech is $\epsilon_k$-differentially private for $\operatorname{Adj}_{d_{\tilde{P}}}^\zeta$ at any time instant $k \in {\mathbb Z}_{k_0 +}$ for any $k_0 \in {\mathbb Z}$ if the system eq:sys is output incrementally bounded and time-varying Laplacian noise eq:noise where $\epsilon_{-1} : = 0.$

Figures (3)

  • Figure 1: Mechanism diagram: The exo-system (fusion center) sends its noisy reference signals to the local system. Information in the green dashed box is not eavesdropped, while information in the red dashed box is vulnerable.
  • Figure 2: Output regulation with system privacy protection.
  • Figure 3: Output Trajectories of Exo-system with Different $\omega$s

Theorems & Definitions (10)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof