Analysis of Boolean Functions (2105.10386v1)

Abstract: The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : {0,1}^n \to {0,1}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps the most basic object of study in theoretical computer science, and Fourier analysis has become an indispensable tool in the field. The topic has also played a key role in several other areas of mathematics, from combinatorics, random graph theory, and statistical physics, to Gaussian geometry, metric/Banach spaces, and social choice theory. The intent of this book is both to develop the foundations of the field and to give a wide (though far from exhaustive) overview of its applications. Each chapter ends with a "highlight" showing the power of analysis of Boolean functions in different subject areas: property testing, social choice, cryptography, circuit complexity, learning theory, pseudorandomness, hardness of approximation, concrete complexity, and random graph theory. The book can be used as a reference for working researchers or as the basis of a one-semester graduate-level course. The author has twice taught such a course at Carnegie Mellon University, attended mainly by graduate students in computer science and mathematics but also by advanced undergraduates, postdocs, and researchers in adjacent fields. In both years most of Chapters 1-5 and 7 were covered, along with parts of Chapters 6, 8, 9, and 11, and some additional material on additive combinatorics. Nearly 500 exercises are provided at the ends of the book's chapters.

• The paper demonstrates that representing Boolean functions as real multilinear polynomials via Fourier expansion enables rigorous analytical study.
• The paper quantifies variable influences using discrete derivatives and total influence, linking sensitivity to noise stability for robust function analysis.
• The paper applies its spectral methods to learning theory and social choice, highlighting efficient learnability and providing a spectral proof of Arrow's Theorem.

Overview of "Analysis of Boolean Functions" by Ryan O'Donnell

This text serves as an extensive examination of Boolean functions through the lens of theoretical computer science and mathematics, particularly focusing on the Fourier analysis of such functions. O'Donnell's work is a significant contribution to the understanding and application of Boolean functions, providing both theoretical foundations and insightful applications across various domains.

Key Concepts and Insights

1. Boolean Functions and Fourier Expansion:
   • A primary focus is the Fourier expansion of Boolean functions, treating them as real multilinear polynomials. This approach is pivotal because it makes numerous properties of Boolean functions analytically tractable.
   • The Fourier transform allows Boolean functions to be expressed in terms of basis functions (parity functions), which significantly facilitates the paper of their structure and properties.
2. Influences and Derivatives:
   • The concept of influence measures how much a specific input variable affects the output of a Boolean function. This notion is deeply connected to the paper of sensitivity and learning theory, providing insights into which variables are most critical to the outcome of a function.
   • The text introduces mathematical tools such as discrete derivatives and Laplacians to formalize and analyze these influences.
3. Noise Stability and Sensitivity:
   • The work explores the effects of noise on Boolean functions, specifically looking at how small perturbations in the input affect the stability of the output. The notion of noise stability is especially critical in applications like learning theory and social choice.
   • O'Donnell provides a connection between noise sensitivity and influence, elaborating on how these concepts can predict the robustness of a Boolean function to random noise.
4. Total Influence and Its Applications:
   • Total influence, or average sensitivity, is shown to be a vital measure for Boolean functions, especially in the context of learning theory where it aids in understanding the complexity of learning these functions from data.
   • The interplay between total influence and edge isoperimetric problems in the Hamming cube is particularly noteworthy, connecting combinatorial geometry with analysis.
5. Spectral Structure and Learning:
   • A thorough investigation into the spectral structure of Boolean functions reveals insights into their complexity and the potential for polynomial-time learnability.
   • O'Donnell highlights how certain classes of functions, when having concentrated spectral weight at low degrees, can be efficiently learned — a significant result for computational learning theory.
6. Arrow's Theorem:
   • Acknowledging classical results from social choice theory, the text provides a spectral proof of Arrow's Theorem, demonstrating the utility of harmonic analysis methods in diverse areas.

Applications and Implications

The analysis has practical implications in various fields. For instance, in computational learning theory, understanding the Fourier coefficients of Boolean functions can lead to efficient algorithms for learning these functions from examples. In social choice, concepts such as influence and noise sensitivity illuminate the properties of voting systems. Theoretical results like the KKL theorem also provide boundary conditions on the extent of influence in large systems, which are crucial for designing robust algorithms and protocols.

O'Donnell's work also opens pathways for future exploration, particularly in improving learning efficiencies and extending these concepts to more complex domains like quantum computing. Moreover, the Fourier analysis framework is primed for advancements in cryptographic functions due to its profound implications in pseudorandomness and secure function evaluation.

In conclusion, "Analysis of Boolean Functions" presents a comprehensive and deeply analytical perspective on Boolean functions, enriched with mathematical rigor and extensive applicability across scientific disciplines. The text not only serves as a scholarly resource but also as a foundation on which further research and application in computer science, mathematics, and beyond can be built.