A Note on the Probability of Rectangles for Correlated Binary Strings

Or Ordentlich; Yury Polyanskiy; Ofer Shayevitz

doi:10.1109/TIT.2020.3018232

A Note on the Probability of Rectangles for Correlated Binary Strings

Or Ordentlich^*, Yury Polyanskiy, Ofer Shayevitz

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.

Original language	American English
Article number	9171899
Pages (from-to)	7878-7886
Number of pages	9
Journal	IEEE Transactions on Information Theory
Volume	66
Issue number	12
DOIs	https://doi.org/10.1109/TIT.2020.3018232
State	Published - Dec 2020

Bibliographical note

Funding Information:
Manuscript received September 3, 2019; revised August 3, 2020; accepted August 10, 2020. Date of publication August 20, 2020; date of current version November 20, 2020. The work of Or Ordentlich was supported by ISF under Grant 1791/17. The work of Yury Polyanskiy was supported in part by the National Science Foundation under Grant CCF-17-17842, in part by the Center for Science of Information (CSoI), and in part by the NSF Science and Technology Center under Grant CCF-09-39370. The work of Ofer Shayevitz was supported by the European Research Council under Grant 639573. (Corresponding author: Or Ordentlich.) Or Ordentlich is with the Rachel and Selim Benin School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: or.ordentlich@mail.huji.ac.il).

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

Isoperimetric inequalities
binary adder multiple access channel (MAC)
hypercontractivity

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10.1109/TIT.2020.3018232

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title = "A Note on the Probability of Rectangles for Correlated Binary Strings",

abstract = "Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.",

keywords = "Isoperimetric inequalities, binary adder multiple access channel (MAC), hypercontractivity",

author = "Or Ordentlich and Yury Polyanskiy and Ofer Shayevitz",

note = "Funding Information: Manuscript received September 3, 2019; revised August 3, 2020; accepted August 10, 2020. Date of publication August 20, 2020; date of current version November 20, 2020. The work of Or Ordentlich was supported by ISF under Grant 1791/17. The work of Yury Polyanskiy was supported in part by the National Science Foundation under Grant CCF-17-17842, in part by the Center for Science of Information (CSoI), and in part by the NSF Science and Technology Center under Grant CCF-09-39370. The work of Ofer Shayevitz was supported by the European Research Council under Grant 639573. (Corresponding author: Or Ordentlich.) Or Ordentlich is with the Rachel and Selim Benin School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: or.ordentlich@mail.huji.ac.il). Publisher Copyright: {\textcopyright} 1963-2012 IEEE.",

year = "2020",

month = dec,

doi = "10.1109/TIT.2020.3018232",

language = "American English",

volume = "66",

pages = "7878--7886",

journal = "IEEE Transactions on Information Theory",

issn = "0018-9448",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

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AU - Ordentlich, Or

AU - Polyanskiy, Yury

AU - Shayevitz, Ofer

N1 - Funding Information: Manuscript received September 3, 2019; revised August 3, 2020; accepted August 10, 2020. Date of publication August 20, 2020; date of current version November 20, 2020. The work of Or Ordentlich was supported by ISF under Grant 1791/17. The work of Yury Polyanskiy was supported in part by the National Science Foundation under Grant CCF-17-17842, in part by the Center for Science of Information (CSoI), and in part by the NSF Science and Technology Center under Grant CCF-09-39370. The work of Ofer Shayevitz was supported by the European Research Council under Grant 639573. (Corresponding author: Or Ordentlich.) Or Ordentlich is with the Rachel and Selim Benin School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: or.ordentlich@mail.huji.ac.il). Publisher Copyright: © 1963-2012 IEEE.

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N2 - Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.

AB - Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.

KW - Isoperimetric inequalities

KW - binary adder multiple access channel (MAC)

KW - hypercontractivity

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JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

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M1 - 9171899

ER -

A Note on the Probability of Rectangles for Correlated Binary Strings

Abstract

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Keywords

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