Related papers: Analysis of Boolean Functions
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods,…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
The goal of the paper is to relate complexity measures associated with the evaluation of Boolean functions (certificate complexity, decision tree complexity) and learning dimensions used to characterize exact learning (teaching dimension,…
This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the…
This book is based on notes compiled over the many years I have been teaching the course "Applied Functional Analysis" in the first year of the Master programme at Delft University of Technology, for students with previous exposure to the…
Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main…
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of…
Items in a test are often used as a basis for making decisions and such tests are therefore required to have good psychometric properties, like unidimensionality. In many cases the sum score is used in combination with a threshold to decide…
Here we present our arguments for a more in-depth study of the Boolean cube, which is one of the most important discrete structures. The article contains a case study that analyses how the Boolean cube has been included and explained in…
This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
We consider the problem of linearizing a pseudo-Boolean function $f : \{0,1\}^n \to \mathbb{R}$ by means of $k$ Boolean functions. Such a linearization yields an integer linear programming formulation with only $k$ auxiliary variables. This…
A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…
This book concentrates on functional analysis. The text is written so that it can be followed on the basis of high school mathematics. The book introduces the set theoretical foundations of mathematics, the basic theories of linear algebra…
About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we offered several proofs of the statement that Boolean functions $f(x_1,x_2,\dots,x_n)$, whose…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…
Apart from an account of classical preliminaries, this volume contains a systematic introduction to Sobolev spaces and functions of bounded variation with selected applications. This is installment III of a four part discussion of certain…
Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown…