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Binomial Channel: On the Capacity-Achieving Distribution and Bounds on the Capacity

Ian Zieder, Antonino Favano, Luca Barletta, Alex Dytso

TL;DR

This paper analyzes the binomial channel with $X\in[0,1]$ and $Y\in\{0,\dots,n\}$, establishing both the structure of the capacity-achieving distribution and firm capacity bounds. It proves uniqueness of the optimal input using total-positivity of the binomial kernel, shows that all optimal inputs are discrete with tight bounds on the support, and improves the cardinality bound to $|\mathrm{supp}(P_{X^*})|\le 2+\lfloor n/2\rfloor$ while giving a $\Theta(\sqrt{n})$ lower bound. The second part derives exact capacity values for $n\le 3$ and provides dual and entropy-based bounds that certify $C(n)$ grows like $\frac{1}{2}\log n$ (with explicit Beta-distribution-based lower bounds and dual upper bounds). Collectively, these results sharpen finite-$n$ and asymptotic insights, inform algorithmic computation of $P_{X^*}$, and deepen understanding of capacity for finite-output, infinite-input channels.

Abstract

This work considers a binomial noise channel. The paper can be roughly divided into two parts. The first part is concerned with the properties of the capacity-achieving distribution. In particular, for the binomial channel, it is not known if the capacity-achieving distribution is unique since the output space is finite (i.e., supported on integers $0, \ldots, n)$ and the input space is infinite (i.e., supported on the interval $[0,1]$), and there are multiple distributions that induce the same output distribution. This paper shows that the capacity-achieving distribution is unique by appealing to the total positivity property of the binomial kernel. In addition, we provide upper and lower bounds on the cardinality of the support of the capacity-achieving distribution. Specifically, an upper bound of order $ \frac{n}{2}$ is shown, which improves on the previous upper bound of order $n$ due to Witsenhausen. Moreover, a lower bound of order $\sqrt{n}$ is shown. Finally, additional information about the locations and probability values of the support points is established. The second part of the paper focuses on deriving upper and lower bounds on capacity. In particular, firm bounds are established for all $n$ that show that the capacity scales as $\frac{1}{2} \log(n)$.

Binomial Channel: On the Capacity-Achieving Distribution and Bounds on the Capacity

TL;DR

This paper analyzes the binomial channel with and , establishing both the structure of the capacity-achieving distribution and firm capacity bounds. It proves uniqueness of the optimal input using total-positivity of the binomial kernel, shows that all optimal inputs are discrete with tight bounds on the support, and improves the cardinality bound to while giving a lower bound. The second part derives exact capacity values for and provides dual and entropy-based bounds that certify grows like (with explicit Beta-distribution-based lower bounds and dual upper bounds). Collectively, these results sharpen finite- and asymptotic insights, inform algorithmic computation of , and deepen understanding of capacity for finite-output, infinite-input channels.

Abstract

This work considers a binomial noise channel. The paper can be roughly divided into two parts. The first part is concerned with the properties of the capacity-achieving distribution. In particular, for the binomial channel, it is not known if the capacity-achieving distribution is unique since the output space is finite (i.e., supported on integers and the input space is infinite (i.e., supported on the interval ), and there are multiple distributions that induce the same output distribution. This paper shows that the capacity-achieving distribution is unique by appealing to the total positivity property of the binomial kernel. In addition, we provide upper and lower bounds on the cardinality of the support of the capacity-achieving distribution. Specifically, an upper bound of order is shown, which improves on the previous upper bound of order due to Witsenhausen. Moreover, a lower bound of order is shown. Finally, additional information about the locations and probability values of the support points is established. The second part of the paper focuses on deriving upper and lower bounds on capacity. In particular, firm bounds are established for all that show that the capacity scales as .
Paper Structure (24 sections, 24 theorems, 126 equations, 1 figure, 1 table)

This paper contains 24 sections, 24 theorems, 126 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Outline and Contributions
  4. Notation
  5. Preliminaries
  6. KKT Conditions
  7. Estimation Theoretic Preliminaries
  8. Properties of the Capacity-Achieving Distributions
  9. Symmetry
  10. Discreteness
  11. Uniqueness of the Optimal Input Distribution
  12. On the Location of Support Points
  13. Bounds on the Probabilities
  14. Bounds on the Cardinality
  15. Capacity and Bounds on the Capacity
  16. ...and 9 more sections

Key Result

Lemma 1

$P_{X^\star}$ is a capacity-achieving input distribution if and only if the following conditions hold: where $P_{X^\star} \to P_{Y|X} \to P_{Y^\star}$ (i.e., the optimal output distribution) and

Figures (1)

  • Figure 1: Comparison of the bounds on the crest-factor $\mathcal{D}(x)$ reported in Proposition \ref{['prop:crest_factor_bounds']} for $n=10$.

Theorems & Definitions (43)

  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • Lemma 3
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • ...and 33 more