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Entropy Functions on Two-Dimensional Faces of Polymatroidal Region of Degree Four: Part I: Problem Formulation and More

Shaocheng Liu, Qi Chen

TL;DR

The paper develops a faces-based framework to characterize entropy functions for the 2D faces of the degree-4 polymatroidal region. It enumerates 59 face types via an algorithm and completely characterizes 49, using matroidal and graph-coloring techniques to handle the non-trivial cases. Part I focuses on extreme-ray analysis and the entropy structure induced on 2D faces, setting up Part II to finish the remaining 10 faces (8 fully and 2 partially) and extending the approach. The work advances the understanding of how entropy functions concentrate on low-dimensional faces of the polymatroidal region and builds a bridge to combinatorial objects like Latin squares and related designs.

Abstract

Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this series of two papers, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region of degree 4. In Part I, we formulate the problem, enumerate all 59 types of 2-dimensional faces of the region by an algorithm, and fully characterize entropy functions on 49 types of them. Among them, those non-trivial cases are mainly characterized by the graph-coloring technique. The entropy functions on the remaining 10 types of faces will be characterized in Part II, among which 8 types are fully characterized, and 2 types are partially characterized.

Entropy Functions on Two-Dimensional Faces of Polymatroidal Region of Degree Four: Part I: Problem Formulation and More

TL;DR

The paper develops a faces-based framework to characterize entropy functions for the 2D faces of the degree-4 polymatroidal region. It enumerates 59 face types via an algorithm and completely characterizes 49, using matroidal and graph-coloring techniques to handle the non-trivial cases. Part I focuses on extreme-ray analysis and the entropy structure induced on 2D faces, setting up Part II to finish the remaining 10 faces (8 fully and 2 partially) and extending the approach. The work advances the understanding of how entropy functions concentrate on low-dimensional faces of the polymatroidal region and builds a bridge to combinatorial objects like Latin squares and related designs.

Abstract

Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this series of two papers, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region of degree 4. In Part I, we formulate the problem, enumerate all 59 types of 2-dimensional faces of the region by an algorithm, and fully characterize entropy functions on 49 types of them. Among them, those non-trivial cases are mainly characterized by the graph-coloring technique. The entropy functions on the remaining 10 types of faces will be characterized in Part II, among which 8 types are fully characterized, and 2 types are partially characterized.
Paper Structure (14 sections, 28 theorems, 185 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 28 theorems, 185 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Faces of a polyhedral cone
  5. Entropy functions on the extreme rays of $\Gamma_4$
  6. Two-dimensional faces of $\Gamma_4$
  7. Characterization of entropy functions on two-dimensional faces of $\Gamma_4$
  8. All-entropic faces
  9. Entropy functions on the faces extended from $2$-dimensional faces of $\Gamma_3$
  10. Entropy functions on the faces involving $U_{2,3}^{123}$
  11. Entropy functions on the faces involving partitions
  12. Entropy functions on the faces involving $U_{3,4}$
  13. Entropy functions on the faces involving $\hat{U}_{3,5}^{1}$
  14. Non-entropic faces

Key Result

Theorem 2.3.1

For $E=U^1_{1,1}, U^{12}_{1,2}, U^{123}_{1,3}, U_{1,4}$, i.e., extreme rays containing a matroid with rank $1$, $a\mathbf{r}\in E$ is entropic for all $a\ge 0$.

Figures (8)

  • Figure 1: The faces $(U_{2,3}^{123},U_{1,2}^{12})$, $(\mathcal{W}_{2}^{34},U_{1,2}^{12})$ and $(\mathcal{W}_{2}^{14},U_{1,3}^{124})$
  • Figure 2: The faces$(U_{2,3}^{123},U_{1,1}^{1}), (U_{2,3}^{123},U_{1,1}^{4}), (U_{2,3}^{123},U_{1,2}^{14}),$$(\mathcal{W}_{2}^{14},U_{1,1}^{1}), (\mathcal{W}_{2}^{34},U_{1,1}^{1}), (\mathcal{W}_{2}^{14},U_{1,2}^{14})$ and $(\mathcal{W}_{2}^{24}, U_{1,2}^{14})$
  • Figure 3: The face $(U_{2,3}^{123} , U_{1,3}^{124})$
  • Figure 4: The face $( U_{2,3}^{123 } , U_{2,3}^{124 } )$
  • Figure 5: The face $( U_{2,3}^{123} , U_{1,4} )$
  • ...and 3 more figures

Theorems & Definitions (57)

  • Theorem 2.3.1
  • proof
  • Theorem 2.3.2
  • Theorem 2.3.3
  • Theorem 2.3.4
  • proof
  • Lemma 1
  • proof
  • Theorem 2.3.5
  • proof
  • ...and 47 more