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On Non-Interactive Simulation of Distributed Sources with Finite Alphabets

Hojat Allah Salehi, Farhad Shirani

TL;DR

A constructive algorithm is introduced that explicitly finds the simulating functions of the simulating functions of the Fourier analysis framework, thus providing a super-exponential improvement in input complexity.

Abstract

This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences $X^d$ and $Y^d$ drawn according to a joint distribution $P_{X^dY^d}$. The agents aim to generate outputs $U=f_d(X^d)$ and $V=g_d(Y^d)$ with a joint distribution sufficiently close in total variation to a target distribution $Q_{UV}$. Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is $O(\exp\operatorname{poly}(\frac{1}ε))$. In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is $Θ(\log{\frac{1}ε})$, thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions $(f_d(X^d),g_d(Y^d))$. The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.

On Non-Interactive Simulation of Distributed Sources with Finite Alphabets

TL;DR

A constructive algorithm is introduced that explicitly finds the simulating functions of the simulating functions of the Fourier analysis framework, thus providing a super-exponential improvement in input complexity.

Abstract

This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences and drawn according to a joint distribution . The agents aim to generate outputs and with a joint distribution sufficiently close in total variation to a target distribution . Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is . In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is , thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions . The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.
Paper Structure (32 sections, 20 theorems, 117 equations, 8 figures, 1 algorithm)

This paper contains 32 sections, 20 theorems, 117 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. The Non-interactive Source Simulation Problem
  4. Discrete Fourier Expansion
  5. A Fourier Framework for Quantifying Distributed Correlation
  6. Functions as Overparametrized Vectors
  7. Fourier Expansion and Disagreement Probability
  8. Non-Convexity of $\mathcal{Q}(P_{XY}, \mathcal{U}, \mathcal{V})$ and Decomposition into Star-Convex Sets
  9. Extended Search Space and the Randomization/Derandomization Procedure
  10. Binary-Output Randomization and Derandomization
  11. Finite-Alphabet-Output Randomization and Derandomization
  12. Expanded Search Space and A Bijective Mapping
  13. The Maximal Correlation Problem and Solvability of NISS
  14. Primal and Dual Formulations of the Maximal Correlation Problem
  15. Case Studies in Classical and Quantum Distributed Correlation Quantification
  16. ...and 17 more sections

Key Result

Lemma 1

Let $P_{XY}$ be a probability measure over $\mathbb{F}_q\times \mathbb{F}_q$. For any bounded pair of functions $f,g: \mathbb{F}_q\mapsto \mathbb{R}$, the following holds where $\rho_{s,t}= \mathbb{E}_{X,Y}[\psi_{s}(X)\psi'_t(Y)], s,t\in \mathbb{F}_q$ is the correlation coefficient between $\psi_{s}(X)$ and $\psi'_t(Y)$ under $P_{XY}$, $(\psi_s, s\in \mathbb{F}_q)$ and $(\psi'_t, t\in \mathbb{F}_

Figures (8)

  • Figure 1: The non-interactive source simulation scenario.
  • Figure 2: An NISS scenario with one-bit common randomness and unlimited local randomness available to the agents.
  • Figure 3: A quantum NISS scenario with a Bell state shared among two agents along with unlimited local randomness available to the agents.
  • Figure 4: An NISS scenario with first-order Markov sources. We have defined $\delta_z\triangleq \delta_x\ast\delta_y$.
  • Figure 5: Achievable biased maximal correlation $\rho_b$ for the symmetric-output BB-NISS using F-PATH as a function of number of input samples $d$ and output marginals $P_U(1)=P_V(1)=P(1)$.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1: NISS
  • Remark 1
  • Definition 2: NISS with threshold $\epsilon$
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Proposition 1
  • Lemma 3: Star-Convexity of $\mathcal{P}(P_{XY},Q_U,Q_V)$
  • Definition 3: Binary-Output Randomized Simulating Functions
  • Definition 4: Finite-Output Randomized Simulating Functions
  • ...and 26 more