Optimal Reconstruction from Linear Queries

2026-05-19

AI Summary

This work investigates the optimal reconstruction of an unknown vector in ℝᵈ from noisy linear queries, characterizing the fundamental trade-offs among reconstruction error, number of queries, ambient dimension, and noise level. By integrating geometric analysis, symmetry considerations, and a dynamical framework based on Lie group actions, the study establishes—for the first time—the optimal error bound in the infinite-query regime and extends Jung’s theorem to robust near-extremal settings. The main results show that the minimal achievable error equals √(2d/(d+1))δ, where δ quantifies the noise magnitude. In fixed dimensions, the excess error decays doubly exponentially with the number of queries, whereas in high-dimensional regimes, an exponential number of queries—on the order of exp(d)—is necessary for the error to vanish asymptotically.

Abstract

We study the problem of reconstructing an unknown point in $\mathbb{R}^d$ from approximate linear queries. This setting arises naturally in applications ranging from low-dimensional remote sensing and signal recovery to high-dimensional data analysis and privacy-sensitive inference. Our main goal is to characterize the optimal reconstruction error as a function of the number of queries $T$, the ambient dimension $d$, and the noise parameter $δ$. We first analyze the limit $T \to \infty$ and show that the optimal reconstruction error converges to the explicit value $\sqrt{2d/(d+1)} δ$, which plays a role analogous to the Bayes optimal error in supervised learning. When the dimension is fixed, we show that the excess error above this limit decays doubly exponentially fast as $T \to \infty$, a rate that is significantly faster than those typically encountered in learning curves. When the dimension grows, we show that a number of queries on the order of $\exp(d)$ is necessary and sufficient to achieve vanishing excess error. Finally, we introduce and analyze an improper variant of the reconstruction problem. From a technical perspective, our main contribution is a generalization of Jung's theorem (1901). The classical theorem bounds the maximum possible radius of a set of diameter 1 and characterizes extremal bodies. Our generalization provides a robust variant that characterizes near-extremal bodies and is proved via geometric and dynamical arguments exploiting symmetry and Lie group actions.

Problem

• linear queries
• reconstruction
• noise
• dimension
• error

Innovation

• Jung's theorem
• robust geometric analysis
• linear queries
• optimal reconstruction error
• Lie group actions