High-Dimensional Signal Compression: Lattice Point Bounds and Metric Entropy

TL;DR

This paper investigates worst-case high-dimensional signal compression under energy constraints, providing explicit bounds on codebook size by analyzing lattice points in ellipsoids, refining classical estimates.

Contribution

It introduces new explicit bounds on codebook size for high-dimensional compression using lattice point counting and refined classical estimates.

Findings

Derived dimension-dependent upper bounds on codebook size.

Refined Landau's lattice point estimates with uniform Bessel bounds.

Applied Abel summation for explicit bounds.

Abstract

We study worst-case signal compression under an $\ell^2$ energy constraint, with coordinate-dependent quantization precisions. The compression problem is reduced to counting lattice points in a diagonal ellipsoid. Under balanced precision profiles, we obtain explicit, dimension-dependent upper bounds on the logarithmic codebook size. The analysis refines Landau's classical lattice point estimates using uniform Bessel bounds due to Olenko and explicit Abel summation.