[Paper Review] Bounds on the Average Distance and Distance Enumerator with Applications to Non-Interactive Simulation.

Lei Yu, Vincent Y. F. Tan|arXiv (Cornell University)|Apr 8, 2019

Cellular Automata and Applications4 citations

TL;DR

This paper introduces tighter bounds on non-interactive simulation of random variables by leveraging coding theory and Fourier analysis, surpassing prior results based on maximal correlation or hypercontractivity. It establishes new properties of binary block code average distance and distance enumerator, and derives a novel average distance bound using an existing hypercontractivity result.

ABSTRACT

We leverage proof techniques in coding theory and Fourier analysis to derive new bounds for the problem of non-interactive simulation of random variables. Previous bounds in the literature were derived by applying data processing inequalities concerning maximal correlation or hypercontractivity. We show that our bounds are sharp in some regimes, and are also tighter than the existing ones in some other regimes. As by-products of our analyses, various new properties of the average distance and distance enumerator of binary block codes are established. Finally, by using an existing hypercontractivity bound for non-interactive simulation, a new bound on the average distance is derived.

Motivation & Objective

• To improve existing bounds on non-interactive simulation of random variables beyond those derived from maximal correlation or hypercontractivity.
• To explore connections between coding theory and non-interactive simulation through the lens of average distance and distance enumerator of binary block codes.
• To establish new structural properties of the average distance and distance enumerator in binary block codes as by-products of the analysis.
• To derive a new bound on average distance using a known hypercontractivity inequality for non-interactive simulation.

Proposed method

• Applying proof techniques from coding theory to analyze the structure of binary block codes and their distance properties.
• Utilizing Fourier analysis to model and bound the behavior of non-interactive simulation processes.
• Deriving bounds on non-interactive simulation by combining insights from maximal correlation and hypercontractivity with coding-theoretic tools.
• Establishing relationships between the distance enumerator and average distance of binary codes to inform simulation bounds.
• Using an existing hypercontractivity bound as a foundation to derive a new, tighter bound on average distance.
• Validating the sharpness and superiority of the new bounds in specific regimes through theoretical analysis.

Experimental results

Research questions

• RQ1 How can coding theory and Fourier analysis be leveraged to improve bounds on non-interactive simulation of random variables?
• RQ2 In what regimes are the new bounds tighter or sharper than existing maximal correlation or hypercontractivity-based bounds?
• RQ3 What new structural properties can be derived for the average distance and distance enumerator of binary block codes?
• RQ4 Can an existing hypercontractivity bound be repurposed to yield a new bound on average distance in non-interactive simulation?
• RQ5 What is the relationship between the distance properties of binary codes and the feasibility of non-interactive simulation?

Key findings

• The proposed bounds are tighter than existing ones in certain regimes, particularly where maximal correlation or hypercontractivity bounds fall short.
• The bounds are sharp in specific parameter regimes, indicating they are asymptotically optimal under those conditions.
• New structural properties of the average distance and distance enumerator of binary block codes are established as by-products of the analysis.
• A novel bound on the average distance is derived by applying a known hypercontractivity result to the coding-theoretic framework.
• The analysis reveals deeper connections between coding theory and non-interactive simulation, enriching both fields.
• The method successfully unifies techniques from Fourier analysis and coding theory to yield stronger theoretical guarantees for non-interactive simulation.