Privacy-Preserving Cram\'er-Rao Lower Bound

TL;DR

This paper develops a theoretical framework for understanding the fundamental limits of identification accuracy under privacy constraints, extending classical bounds to multi-sensor systems and proposing algorithms to achieve these bounds.

Contribution

It introduces the privacy-preserving Cramér-Rao lower bound theory, including an identifiability criterion, and extends it to multi-sensor systems with distributed algorithms.

Findings

Established the privacy-preserving CR lower bound and its attainability.

Proved additivity of privacy-preserving Fisher information in multi-sensor systems.

Demonstrated effectiveness of distributed algorithms achieving the bounds.

Abstract

This paper establishes the privacy-preserving Cram\'er-Rao (CR) lower bound theory, characterizing the fundamental limit of identification accuracy under privacy constraint. An identifiability criterion under privacy constraint is derived by using Fisher information matrix as the privacy metric. In the identifiable case, the privacy-preserving CR lower bound is established and its attainability is demonstrated, thereby ensuring the existence of the privacy-preserving Fisher information matrix with explicit expression. Then, the privacy-preserving CR lower bound theory is extended to the multi-sensor multi-measurement system. Specifically, the additivity principle of privacy-preserving Fisher information matrices across both spatial and temporal dimensions is established, building a relationship between privacy-preserving CR lower bounds for the multi-sensor multi-measurement system and its subsystems. Using this additivity principle, distributed identification algorithms capable of achieving the privacy-preserving CR lower bound are further proposed. Numerical examples are provided to demonstrate the privacy-preserving CR lower bound and show the effectiveness of the proposed algorithms.