Reconstruction and Secrecy under Approximate Distance Queries
This work addresses privacy and accuracy trade-offs in localization and data access, offering theoretical insights with potential applications in GPS, sensor networks, and security, but it is incremental as it builds on classical geometric concepts.
The paper tackles the problem of locating an unknown target using noisy distance queries, analyzing the optimal reconstruction error and its asymptotic behavior. It provides a tight geometric characterization of this error in terms of the Chebyshev radius for compact metric spaces and characterizes pseudo-finiteness for convex Euclidean spaces.
Consider the task of locating an unknown target point using approximate distance queries: in each round, a reconstructor selects a query point and receives a noisy version of its distance to the target. This problem arises naturally in various contexts ranging from localization in GPS and sensor networks to privacy-aware data access, and spans a wide variety of metric spaces. It is relevant from the perspective of both the reconstructor (seeking accurate recovery) and the responder (aiming to limit information disclosure, e.g., for privacy or security reasons). We study this reconstruction game through a learning-theoretic lens, focusing on the rate and limits of the best possible reconstruction error. Our first result provides a tight geometric characterization of the optimal error in terms of the Chebyshev radius, a classical concept from geometry. This characterization applies to all compact metric spaces (in fact, even to all totally bounded spaces) and yields explicit formulas for natural metric spaces. Our second result addresses the asymptotic behavior of reconstruction, distinguishing between pseudo-finite spaces -- where the optimal error is attained after finitely many queries -- and spaces where the approximation curve exhibits nontrivial decay. We characterize pseudo-finiteness for convex Euclidean spaces.