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Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity

Lei Yu, Venkat Anantharam, Jun Chen

TL;DR

The paper analyzes graphs formed by joint-type pairs, deriving exact asymptotic descriptions for the maximal edge-density exponent via E^*(R_1,R_2) and the asymptotic biclique rate region via \\mathcal{R}(T_{XY}) = \\mathcal{R}^*(T_{XY}). It leverages the method of types and linear-algebraic tools to connect these combinatorial structures to noninteractive simulation and to Brascamp–Lieb/hypercontractivity inequalities, establishing stronger bounds and explicit regions for finite alphabets, with DSBS as a principal example. A key application yields a sharper outer bound for the zero-error capacity region of the binary adder channel. The work also extends classical inequalities to small-support nonnegative functions, providing exponentially sharp forward and reverse hypercontractivity results and clarifying the role of type overflow in exact versus approximate regions. Overall, the results bridge combinatorial type-graph properties with information-theoretic notions and have potential impact on zero-error coding and hypercontractivity theory.

Abstract

In this paper, we study the type graph, namely, a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length $n$ of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of $(R_{1},R_{2})$ such that there exists a biclique of the type graph which has respectively $2^{nR_{1}}$ and $2^{nR_{2}}$ vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then discuss the connections of these results to noninteractive simulation and hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types and linear algebra.

Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity

TL;DR

The paper analyzes graphs formed by joint-type pairs, deriving exact asymptotic descriptions for the maximal edge-density exponent via E^*(R_1,R_2) and the asymptotic biclique rate region via \\mathcal{R}(T_{XY}) = \\mathcal{R}^*(T_{XY}). It leverages the method of types and linear-algebraic tools to connect these combinatorial structures to noninteractive simulation and to Brascamp–Lieb/hypercontractivity inequalities, establishing stronger bounds and explicit regions for finite alphabets, with DSBS as a principal example. A key application yields a sharper outer bound for the zero-error capacity region of the binary adder channel. The work also extends classical inequalities to small-support nonnegative functions, providing exponentially sharp forward and reverse hypercontractivity results and clarifying the role of type overflow in exact versus approximate regions. Overall, the results bridge combinatorial type-graph properties with information-theoretic notions and have potential impact on zero-error coding and hypercontractivity theory.

Abstract

In this paper, we study the type graph, namely, a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of such that there exists a biclique of the type graph which has respectively and vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then discuss the connections of these results to noninteractive simulation and hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types and linear algebra.

Paper Structure

This paper contains 22 sections, 18 theorems, 144 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivations
  3. Related Works
  4. Main Contributions
  5. Notation
  6. Type Graphs
  7. Exponents
  8. Biclique Rate Region
  9. Noninteractive Simulation
  10. Sources $\mathrm{Unif}(\mathcal{T}_{T_{XY}})$
  11. Sources $P_{XY}^{n}$
  12. Example: DSBS
  13. Applications to Zero-Error Coding
  14. Brascamp--Lieb and Hypercontractivity Inequalities
  15. Concluding Remarks
  16. ...and 7 more sections

Key Result

Theorem 1

Given a joint $n$-type $T_{XY}$ with $n\ge2(|\mathcal{X}||\mathcal{Y}|+2)|\mathcal{X}||\mathcal{Y}|$, for $(R_{1},R_{2})\in\mathcal{R}_{X}^{(n)}\times\mathcal{R}_{Y}^{(n)}$, we have where $\varepsilon_{n}:=\frac{(|\mathcal{X}||\mathcal{Y}|+2)|\mathcal{X}||\mathcal{Y}|}{n}\log\frac{(n+1)n^{6}}{|\mathcal{X}|^{4}|\mathcal{Y}|^{4}}$. As a consequence, for any $n\ge1$ and any joint $n$-type $T_{XY}$,

Figures (2)

  • Figure 1: Illustration of $\underline{\Upsilon}^{*}$ for $\rho=0.9$.
  • Figure 2: Illustration of $\varphi(s,t)$, $\psi(s,t)$, $\underline{\Theta}^{*}(E_{1},E_{2})$, and $\overline{\Theta}^{*}(E_{1},E_{2})$ for the DSBS for $\rho=0.9$. Note that $\underline{\Theta}^{*}(E_{1},E_{2})$ and $\overline{\Theta}^{*}(E_{1},E_{2})$ are expressed in terms of $\varphi(s,t)$ and $\psi(s,t)$ in (\ref{['eq:lce']}) and (\ref{['eq:uce']}). All the bases of logarithms are 2 for these figures.

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Corollary 1
  • Theorem 2
  • Remark 4
  • Proposition 1
  • Proposition 2
  • ...and 24 more