Optimal Reconstruction from Linear Queries

TL;DR

This paper characterizes the optimal reconstruction error for an unknown point in high-dimensional space from approximate linear queries, revealing exponential query complexity and a generalized geometric theorem.

Contribution

It introduces a generalized Jung's theorem and analyzes the asymptotic behavior of reconstruction error in high-dimensional settings.

Findings

Optimal error converges to rac{2d}{d+1}rac{elta}

Excess error decays doubly exponentially with the number of queries for fixed dimension

xp(d) queries are needed for vanishing error in high dimensions

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 $\delta$. We first analyze the limit $T \to \infty$ and show that the optimal reconstruction error converges to the explicit value $\sqrt{2d/(d+1)} \delta$, 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.