Limits of Discrete Energy of Families of Increasing Sets

Hari Sarang Nathan

math.CAApr 15, 2025

Limits of Discrete Energy of Families of Increasing Sets

Authors: Hari Sarang Nathan

Abstract

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$ , ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$ . We also discuss applications to data science and Erdős/Falconer type problems.

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Optimal Reconstruction from Linear Queries

Yuval Filmus, Shay Moran, Elizaveta Nesterova

Technion – Israel Institute of Technology · Google Research

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.

Reconstruction Error Optimal Sample Complexity Low-Dimensional Structure Spectral Norm Remote Sensing

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Abstract

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