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

A. Iosevich, A. Vagharshakyan, E. Wyman

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.

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

Abstract

We study worst-case signal compression under an 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.

Paper Structure

This paper contains 14 sections, 7 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Compression
  3. The class $\mathcal{X}$
  4. Problem setting
  5. The compression procedure
  6. The recovery procedure
  7. The compression problem
  8. Reduction to a number theory problem
  9. Relation to counting lattice points and metric entropy
  10. Formula (38) of EL
  11. A rough entropy estimate
  12. Abel summation
  13. Olenko's uniform estimate
  14. The $\Delta$ operator

Key Result

Theorem 2.4

Assume $k\geq 2$ and $R\geq 2$. Then where $O(1)$ is bounded as $k\to\infty$. If, in addition, Condition cond:balanced holds, then $\varepsilon_{\mathrm{geom}}$ is comparable to $\varepsilon_{\mathrm{total}}/k$, and mainbound can be rewritten as The implicit constants may depend on the constant $C$ in Condition cond:balanced. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (15)

  • Definition 2.1: Ellipsoidal codebook size
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5: Covering number interpretation
  • Theorem 6.1: Olenko
  • Lemma 6.2
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof
  • ...and 5 more