Optimal Reconstruction from Linear Queries

Yuval Filmus, Shay Moran, Elizaveta Nesterova
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:2428-2476, 2026.

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 \emph{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 \emph{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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-filmus26a, title = {Optimal Reconstruction from Linear Queries}, author = {Filmus, Yuval and Moran, Shay and Nesterova, Elizaveta}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {2428--2476}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/filmus26a/filmus26a.pdf}, url = {https://proceedings.mlr.press/v336/filmus26a.html}, 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 \emph{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 \emph{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.} }
Endnote
%0 Conference Paper %T Optimal Reconstruction from Linear Queries %A Yuval Filmus %A Shay Moran %A Elizaveta Nesterova %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-filmus26a %I PMLR %P 2428--2476 %U https://proceedings.mlr.press/v336/filmus26a.html %V 336 %X 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 \emph{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 \emph{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.
APA
Filmus, Y., Moran, S. & Nesterova, E.. (2026). Optimal Reconstruction from Linear Queries. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:2428-2476 Available from https://proceedings.mlr.press/v336/filmus26a.html.

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