Analysis of Boolean Functions: List of Notation
Ryan O'Donnell
- 01 Jan 2014
775
TL;DR: Boolean functions are perhaps the most basic objects of study in theoretical computer science.
read more
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Posted Content
New lower bounds for Massively Parallel Computation from query complexity
TL;DR: It is proved that any adaptive algorithm for the 1v2-Cycle problem with I/O capacity $O(n^{\varepsilon})$ per machine requires $\Omega(1/\varpsilon)$ rounds, matching a recent upper bound of Behnezhad et al.
2
•Posted Content
Convergence and Sample Complexity of SGD in GANs.
TL;DR: This work considers learning a target distribution modeled by a 1-layer Generator network with a non-linear activation function $\phi(\cdot)$ parametrized by a weight matrix and can learn the target distribution within total-variation distance $\epsilon$ using $\tilde O(d^2/\ep silon^2)$ samples which is (near-)information theoretically optimal.
2
Low-Complexity Weak Pseudorandom Functions in \(\mathtt {AC}0[\mathtt {MOD}2]\)
Elette Boyle,Geoffroy Couteau,Niv Gilboa,Yuval Ishai,Lisa Kohl,Peter Scholl +5 more
- 16 Aug 2021
TL;DR: Weak pseudorandom functions (WPRF) as mentioned in this paper are a keyed function that can be efficiently distinguished from totally random input-output pairs (x, y) under a random key k.
2
•Posted Content
Stability of Equilibrium Asset Pricing Models: A Necessary and Sufficient Condition
TL;DR: The condition is obtained, and hence the problem of existence and uniqueness of asset prices, with the recent literature on stochastic discount factor decompositions, providing a Monte Carlo method that is naturally parallelizable.
2
Lower Bounds of Functions on Finite Abelian Groups
Jianting Yang,Ke Yao,Lihong Zhi +2 more
TL;DR: An algorithm for computing verifiable lower bounds of functions on finite abelian groups is presented. The algorithm utilizes the technique of the Fourier sum of squares with error and a new rounding method to obtain feasible solutions. Experiments on MAX-SAT benchmark problems and random functions showcase the effectiveness of the algorithm.
2
References
•Book
The Theory of Error-Correcting Codes
F.J. MacWilliams,Neil J. A. Sloane +1 more
- 01 Jan 1977
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
12.4K
Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen
TL;DR: In el marco del Proyecto subvencionado by the Fundación Antorchas (FAN) as discussed by the authors, el material was digitalizado, e.g., en la Biblioteca del Departamento de Fisica de la Facultad de Ciencias Exactas de la Universidad Nacional de La Plata.
The Byzantine Generals Problem
TL;DR: The Albanian Generals Problem as mentioned in this paper is a generalization of Dijkstra's dining philosophers problem, where two generals have to come to a common agreement on whether to attack or retreat, but can communicate only by sending messengers who might never arrive.
•Book
Functions of Bounded Variation and Free Discontinuity Problems
Luigi Ambrosio,Diego Pallara,Nicola Fusco +2 more
- 01 Jan 2000
TL;DR: The Mumford-Shah functional minimiser of free continuity problems as mentioned in this paper is a special function of the Mumfordshah functional and has been shown to be a function of free discontinuity set.
5.3K