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Information Properties of a Random Variable Decomposition through Lattices

Fábio C. C. Meneghetti, Henrique K. Miyamoto, Sueli I. R. Costa

TL;DR

This work introduces a decomposition of a real-valued random variable into lattice-wrapping and lattice-quantization components, with ${X} = {X}_\text{π} + {X}_\mathcal{Q}$ defined via a full-rank lattice $\\Lambda$ and a fundamental domain $\\mathcal{D}$. It analyzes information-theoretic quantities such as mutual information $I({X}_\text{π}; {X}_\mathcal{Q})$ and the Fisher information matrix, deriving relations like $I({X}_\text{π}; {X}_\mathcal{Q}) = h({X}_\text{π}) + H({X}_\mathcal{Q}) - h({X})$ and establishing bounds that connect wrapped/quantized entropies to the original entropy. The paper shows that, while the two components are generally dependent, certain memoryless cases yield independence and explores how the information measures behave under scaling of the lattice. A notable contribution is the generalization to topological groups using the Weil formula and Haar measure, enabling a broader information-theoretic treatment of wrapping and quantization beyond Euclidean space. This framework provides a unified perspective for lattice-based coding, shaping, and positional quantization, with potential applications in communications, cryptography, and directional statistics, and sets the stage for further multidimensional and group-theoretic developments.

Abstract

A full-rank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on $\mathbb{R}^n$, two operations can be induced by considering the quotient of the space by such a lattice: wrapping and quantization. For a lattice $Λ$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through $Λ$, the wrapped distribution over the quotient is obtained by summing the density over each coset, while the quantized distribution over the lattice is defined by integrating over each fundamental domain translation. These operations define wrapped and quantized random variables over $D$ and $Λ$, respectively, which sum up to the original random variable. We investigate information-theoretic properties of this decomposition, such as entropy, mutual information and the Fisher information matrix, and show that it naturally generalizes to the more abstract context of locally compact topological groups.

Information Properties of a Random Variable Decomposition through Lattices

TL;DR

This work introduces a decomposition of a real-valued random variable into lattice-wrapping and lattice-quantization components, with defined via a full-rank lattice and a fundamental domain . It analyzes information-theoretic quantities such as mutual information and the Fisher information matrix, deriving relations like and establishing bounds that connect wrapped/quantized entropies to the original entropy. The paper shows that, while the two components are generally dependent, certain memoryless cases yield independence and explores how the information measures behave under scaling of the lattice. A notable contribution is the generalization to topological groups using the Weil formula and Haar measure, enabling a broader information-theoretic treatment of wrapping and quantization beyond Euclidean space. This framework provides a unified perspective for lattice-based coding, shaping, and positional quantization, with potential applications in communications, cryptography, and directional statistics, and sets the stage for further multidimensional and group-theoretic developments.

Abstract

A full-rank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on , two operations can be induced by considering the quotient of the space by such a lattice: wrapping and quantization. For a lattice , and a fundamental domain which tiles through , the wrapped distribution over the quotient is obtained by summing the density over each coset, while the quantized distribution over the lattice is defined by integrating over each fundamental domain translation. These operations define wrapped and quantized random variables over and , respectively, which sum up to the original random variable. We investigate information-theoretic properties of this decomposition, such as entropy, mutual information and the Fisher information matrix, and show that it naturally generalizes to the more abstract context of locally compact topological groups.
Paper Structure (9 sections, 5 theorems, 14 equations, 2 figures)

This paper contains 9 sections, 5 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lattices, Wrapping and Quantization
  3. Lattices and Fundamental Domains
  4. Wrapping and Quantization
  5. Information Properties
  6. Information-theoretic Measures
  7. Fisher Information
  8. A Generalization to Topological Groups
  9. Conclusion

Key Result

Proposition 1

Let ${X}$ be a random variable, and ${X}_\pi$ and ${X}_\mathcal{Q}$ the respective wrapped and quantized random variables, using the lattice $\alpha\mathbb{Z}$. Denote $\mu_\mathcal{Q} \coloneqq \mathop{\mathrm{E}}\nolimits[{X}_\mathcal{Q}]$ and $\sigma_\mathcal{Q}^2 \coloneqq \mathop{\mathrm{Var}}\ If ${X}$ has support $\mathbb{R}$, then $\mathop{\mathrm{I}}\nolimits({X}_\pi; {X}_\mathcal{Q})$ is

Figures (2)

  • Figure 1: Example of zero-mean Gaussian distributions and their corresponding wrapped, quantized and product distributions, with $\Lambda = \mathbb{Z}$ and $\mathcal{D} = [-\frac{1}{2}, \frac{1}{2}[$ for different variances: $\sigma^2 = 0.25$ (blue), $\sigma^2 = 1$ (orange), $\sigma^2 = 4$ (green).
  • Figure 2: Mutual information $\mathop{\mathrm{I}}\nolimits({X}_\pi; {X}_\mathcal{Q})$ and its upper bound.

Theorems & Definitions (18)

  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Example 3
  • Example 4
  • ...and 8 more