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Reconstruction and Secrecy under Approximate Distance Queries

Shay Moran, Elizaveta Nesterova

TL;DR

This work analyzes locating a hidden point in a metric space using noisy distance queries with multiplicative error $\\epsilon$ and additive error $\\delta$. It provides a tight geometric characterization of the asymptotic reconstruction error via the Chebyshev radius, showing $\\mathrm{OPT}_X(\\epsilon,\\delta) = \\mathtt{e}_X((2+\\epsilon)\\delta)$ for totally bounded spaces, and introduces a diameter-radius profile $\\mathtt{e}_X(\\alpha)$ with simple bounds. It also establishes a sharp pseudo-finiteness dichotomy for convex Euclidean spaces: a bounded convex $X \subset \mathbb{R}^n$ is $(\\epsilon,\\delta)$-pseudo-finite iff $\\dim(X)=1$ and $\\epsilon=0$; otherwise not for small $\\delta$, with exponential or double-exponential convergence rates in the number of queries. The results connect to sequential metric dimension and privacy, clarifying how geometry constrains information leakage and the limits of reconstruction under noise, and they provide constructive strategies (translation and rotation) to maintain a surviving feasible region. Overall, the paper advances a geometry-driven understanding of reconstruction under noisy access, with implications for localization, privacy-preserving data analysis, and learning theory.

Abstract

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.

Reconstruction and Secrecy under Approximate Distance Queries

TL;DR

This work analyzes locating a hidden point in a metric space using noisy distance queries with multiplicative error and additive error . It provides a tight geometric characterization of the asymptotic reconstruction error via the Chebyshev radius, showing for totally bounded spaces, and introduces a diameter-radius profile with simple bounds. It also establishes a sharp pseudo-finiteness dichotomy for convex Euclidean spaces: a bounded convex is -pseudo-finite iff and ; otherwise not for small , with exponential or double-exponential convergence rates in the number of queries. The results connect to sequential metric dimension and privacy, clarifying how geometry constrains information leakage and the limits of reconstruction under noise, and they provide constructive strategies (translation and rotation) to maintain a surviving feasible region. Overall, the paper advances a geometry-driven understanding of reconstruction under noisy access, with implications for localization, privacy-preserving data analysis, and learning theory.

Abstract

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.

Paper Structure

This paper contains 35 sections, 14 theorems, 216 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Limit of optimal reconstruction (Theorem \ref{['thm:Jung']}).
  3. Pseudo-finite Spaces (Theorem \ref{['thm:pseudo-finiteness']}).
  4. Problem Setup and Main Results
  5. A Priori vs. A Posteriori Responder.
  6. Optimal Reconstruction Distance
  7. Chebyshev Radius.
  8. Excess Reconstruction Error
  9. Organization.
  10. Examples
  11. Total Boundedness in Theorem \ref{['thm:Jung']}
  12. Pseudo-Finiteness
  13. Technical Overview
  14. Proof of Theorem \ref{['thm:Jung']}
  15. Upper Bound.
  16. ...and 20 more sections

Key Result

Theorem 2

Let $X$ be a totally bounded metric space. Then, for any $\epsilon, \delta \ge 0$, Moreover, if the distance $(2+\epsilon)\delta$ is realized in $X$, i.e., there exists a pair of points at this distance, then

Figures (8)

  • Figure 1: Feasible region $\Phi$ (blue) of the idealized case
  • Figure 2: Feasible region $\Phi$ (blue) of the finite interaction
  • Figure 3: $\Phi(q,r^{\min})$ (blue)
  • Figure 4: $\Phi(q,r^{\max})$ (orange)
  • Figure 5: Visualization of two cases
  • ...and 3 more figures

Theorems & Definitions (50)

  • Example 1: From counting queries to distance queries
  • Remark
  • Theorem 2
  • Definition 4: Pseudo‑finite Spaces
  • Proposition 5: Pseudo-finiteness of the real line
  • Theorem 6
  • Remark
  • Example 7: Unbounded Space: The Real Line
  • Example 8: Bounded but Not Totally Bounded: Discrete Countable Space
  • Example 9: Sparse Subsets of the Real Line
  • ...and 40 more