Fourier analysis of operations that preserve permutations

A revised and expanded second edition of Reiter's classic text Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968). It deals with various developments in analysis centring around the fundamental work of Wiener, Carleman, and especially A. Weil. It starts with the classical theory of Fourier transforms in euclidean space, continues with a study of certain general function algebras, and then discusses functions defined on locally compact groups. The aim is, firstly, to bring out clearly the relations between classical analysis and group theory , and secondly, to study basic properties of functions on abelian and non-abelian groups. The book gives a systematic introduction to these topics and endeavours to provide tools for further research. In the new edition relevant material is added that was not yet available at the time of the first edition.

Abstract: Polynomial factorization is a classical algorithmic algebra problem with a wide range of applications. Of particular interest is factorization over finite fields, among which fields of order two are probably the most important ones when representing Boolean functions by Zhegalkin polynomials. In particular, factorization of Boolean polynomials corresponds to conjunctive decomposition of Boolean functions given in algebraic normal form. In addition, factorization enables decomposition of functions given in full disjunctive normal form (DNF) and positive DNF, as well as Cartesian decomposition of relational data. These applications demonstrate the importance of developing fast factorization algorithms. In this paper, we consider some recently proposed factorization algorithms of polynomial complexity and describe a parallel MIMD implementation that takes advantage of both task-level and data-level parallelism. We conduct some experiments on logic synthesis benchmarks and synthetic (random) polynomials to demonstrate significant factorization speedup. In conclusion, we discuss results of testing a parallel implementation of the algorithm on a massively parallel multicore architecture (REDEFINE). © 2021, Pleiades Publishing, Ltd.

Graph Signal Processing (GSP) is a mathematical framework that aims at extending classical Fourier harmonic analysis to irregular domains described using graphs. Within this framework, authors have proposed to define operators (e.g. translations, convolutions) and processes (e.g. filtering, sampling). A very important and fundamental result in classical harmonic analysis is the uncertainty principle, which states that a signal cannot be localized both in time and in frequency domains. In this chapter, we explore the uncertainty principle in the context of GSP. More precisely, we present notions of graph and spectral spreads, and show that the existence of signals that are both localized in the graph domain and in the spectrum domain depends on the graph.

DCNNs have revolutionized machine learning (ML) and deep learning (DL), achieving remarkable success in image recognition, natural language processing, and other domains. However, their deployment in security-critical industries such as cybersecurity, energy, finance, and the military faces persistent safety and reliability issues. This thesis addresses three primary challenges: the absence of global and precise interpretability in DCNNs, the need for exact, comprehensive, sound, and efficient formal verification of DCNN properties, and the looming privacy threat to user input in cloud-based DCNN deployments for sensitive data. Our research aims to enhance the transparency, reliability, and security of DCNNs, facilitating their wider adoption in critical real-world applications.
\n
\nThis thesis first investigates the potential synergy between cryptanalysis and AI, leveraging deep neural networks (DNNs) to create an automatic cryptographic distinguisher. However, the resulting neural distinguisher (ND) inherits the opacity of DNNs, prompting us to address the “black-box” challenge. Our effort to tackle this very important explainability problem on a mathematically well-defined and small binary size input problem led to the design of methods never before invented in the field of ML/DL.
\n
\nOur approach is inspired by the principles of cipher design. In SPN cipher design, a small cryptographic component known as an S-box (whose input/output sizes usually range from 4 bits to 8 bits) plays a pivotal role in achieving efficient implementation and amenable security analysis, enabling the approximation of large random permutation functions with security guarantees. Building upon this concept, our thesis explores the application of similar principles to the DL field.
\n
\nOur specific aim is to design and develop a compact and small learnable filter for convolutional neural networks (CNNs), with bit widths ranging from 9-bit to 1-bit or 16-bit to 1-bit. When these small filters are assembled into layers and layers into DCNNs, our goal is to enable the efficient approximation of large non-random functions, such as image classifiers. Overall, we managed to reduce the size of elementary building blocks within CNN architectures, making them as analyzable as encryption blocks (S-boxes) for both analytical and implementation purposes. This study is designed to harmonize cryptanalysis and ML/DL by reconceptualizing bulky CNN filter functions in DL into the streamlined, S-box-like structures favored in cryptanalysis.
\n
\nThe innovation lies in creating a dual-purpose neural network operator: one that ML/DL engineers can efficiently train on diverse datasets and subsequently, after training, convert into a format security experts can intuitively scrutinize and fortify.
\n
\nOur approach introduces a novel convolution filter neural network operator, known as the Learnable Truth Table (LTT) filter. First and foremost, our LTT filter is designed to be (i) fully exhaustible in practical time, (ii) learnable, and (iii) easily integrable into an LTT layer and LTT layers easily integrable into a network which we call TTNet. Second, TTNet needs to be scalable on large-scale input instances and complex tasks, enhancing its usefulness in real-world applications. These rules enable the LTT filter to be differentiable while also allowing for the computation of its complete distribution in just \\(2^{16}\\) operations before deployment. The versatility of this LTT filter lies in its ability to be converted, after training, into various forms, including a truth table, conjunctive normal form (CNF), disjunctive normal form (DNF), algebraic normal form (ANF), optimal implementation in Boolean gates (optimal if the computation of its complete distribution can be done in less than \\(2^{12}\\) operations), Boolean rule, and reduced ordered binary decision diagram (ROBDD). This flexibility allows the entire TTNet DCNN to be transformed into an interconnected truth table network, CNFs/DNFs/ANFs, an optimal Boolean gate circuit, a rule-based model, or a sum of ROBDDs.
\n
\nIn the second part of the thesis, we show that the family of Truth Table nets offers an efficient solution to the challenge of analyzing DCNNs. The LTT filter bridges traditional symbolic AI and new learning AI trends. Indeed, symbolic AI is known for being rule-based, explainable, compact, and proven, with the truth table being a standard tool. On the other hand, learning AI is known for being automatable and scalable on complex tasks, with the learnable CNN filter being a standard tool. The LTT filter is a differentiable logic tool that combines the strengths of both AI approaches.
\n
\nThis research extends the application of TTNet to various domains, demonstrating its scalability and effectiveness, notably in classifying large-scale image datasets such as ImageNet. Furthermore, TTNet stands out for its compactness among state-of-the-art differentiable logic gate DNNs, offering an efficient solution for Boolean logic gate circuits conversion of DNNs. Moving beyond foundational capabilities, we have developed four distinct variations of TTNet (TTVerif, TTRules, TT-SCA, TT-FHE) to address specific challenges across four applications. TTVerif shows our network's ability to perform formal, exact, complete, and sound verification of adversarial attack robustness properties across various noise levels. TTRules applies TTNet's principles to complex tabular data from financial, genetic, and healthcare sectors, aiming for global exact interpretability. In the realm of side-channel attacks (SCA), TT-SCA elucidates DNN behavior, identifying critical points of masking and leakage. Lastly, TT-FHE evidences TTNet's potential to secure user data privacy in cloud deployments through fully homomorphic encryption (FHE). These adaptations underscore TTNet's broad applicability and its capacity to solve multifaceted problems in and beyond cryptography.
\n
\nTo conclude, this thesis demonstrates the applicability of TTNet to image classification tasks, mono-channel time series classification tasks, tabular datasets classification and regression tasks, and various critical security applications. TTNet uniquely combines compactness, verification, interpretability, and privacy preservation. The thesis aimed to develop a versatile and robust neural network operator (LTT filter) that bridges symbolic AI and learning AI, facilitating deployments in environments where safety is paramount, thereby rendering it a valuable tool for regulation.

The growing size of modern datasets necessitates splitting a large scale computation into smaller computations and operate in a distributed manner. Adversaries in a distributed system deliberately send erroneous data in order to affect the computation for their benefit. Boolean functions are the key components of many applications, e.g., verification functions in blockchain systems and design of cryptographic algorithms. We consider the problem of computing a Boolean function in a distributed computing system with particular focus on security against Byzantine workers. Any Boolean function can be modeled as a multivariate polynomial with high degree in general. However, the security threshold (i.e., the maximum number of adversarial workers can be tolerated such that the correct results can be obtained) provided by the recent proposed Lagrange Coded Computing (LCC) can be extremely low if the degree of the polynomial is high. We propose three different schemes called coded Algebraic normal form (ANF), coded Disjunctive normal form (DNF) and coded polynomial threshold function (PTF). The key idea of the proposed schemes is to model it as the concatenation of some low-degree polynomials and threshold functions. In terms of the security threshold, we show that the proposed coded ANF and coded DNF are optimal by providing a matching outer bound.

Harmonic analysis plays an essential role in understanding a host of engineering, mathematical, and scientific ideas. In Harmonic Analysis and Applications, the analysis and synthesis of functions in terms of harmonics is presented in such a way as to demonstrate the vitality, power, elegance, usefulness, and the intricacy and simplicity of the subject. This book is about classical harmonic analysis - a textbook suitable for students, and an essay and general reference suitable for mathematicians, physicists, and others who use harmonic analysis.Throughout the book, material is provided for an upper level undergraduate course in harmonic analysis and some of its applications. In addition, the advanced material in Harmonic Analysis and Applications is well-suited for graduate courses. The course is outlined in Prologue I. This course material is excellent, not only for students, but also for scientists, mathematicians, and engineers as a general reference. Chapter 1 covers the Fourier analysis of integrable and square integrable (finite energy) functions on R. Chapter 2 of the text covers distribution theory, emphasizing the theory's useful vantage point for dealing with problems and general concepts from engineering, physics, and mathematics. Chapter 3 deals with Fourier series, including the Fourier analysis of finite and infinite sequences, as well as functions defined on finite intervals. The mathematical presentation, insightful perspectives, and numerous well-chosen examples and exercises in Harmonic Analysis and Applications make this book well worth having in your collection.

Introduction. The purpose of this article is to introduce the general mathematical community to some recent developments in algebraic geometry and nonarchimedean analysis. Let r = p,p a rational prime. Then these developments center around the beginnings of an theory of the polynomial ring ¥r[T] over the finite field of r elements. The goal of this theory is to use nonarchimedean analysis to do for Yr[T] what classical analysis does for Z. The theory allows us to find direct analogues of many of the classical functions of arithmetic interest in a situation that, at first glance, seems as nonclassical as possible. In the process much will be learned about the polynomials. Much also will be learned about the unique properties of Z and the classical functions. One of the exciting aspects of the theory is its great generality. Indeed, we could replace ¥r[T] with much more general affine rings of curves over finite fields. More precisely, if C is a projective, smooth curve over Fr, oo a rational point and A the functions regular away from oo, then we may use A instead of ¥r[T]. Thus one can, so to speak, get a sense of what analysis might have been forced to if Z were not a unique factorization domain. Such observations can only come in the present setting since Q is the only totally real (i.e., all Galois conjugates contained in R) field with a unique absolute-value. We have chosen to stick to the polynomials in order to keep the exposition as simple as possible. The jump from the polynomials to more general rings is not terribly large and most essential features appear for Fr[T]. Another exciting aspect is that we begin to see how a given 'arithmetic* situation generates an associated harmonic analysis. As classical harmonic analysis is based on the integers, the one developed here is based on Fr[T]. In contrast to classical harmonic analysis which is multiplicative, i.e., based on the exponential function, the one here is based on addition. Throughout the paper we compare the theory here with the classical one. In this fashion we hope the reader may speedily develop a feel for the subject. One of the most surprising (and hotly contested) aspects of classical analysis is its harmonic analysis. This centered around the possibility of expanding an arbitrary singly-periodic function in terms of sines and cosines. Since sines and cosines are easily expressed in terms of the exponential function, the central role of this function is apparent. Viewed on the complex plane, e has the following very well-known properties: It is never zero, takes addition to multiplication, is invariant under z H> z + 2777 and, finally, it is its own derivative. As a consequence, e gives

A modified tidal harmonic analysis model for short-term water level observations

We study the complexity of approximating Boolean functions with disjunctive normal forms (DNFs) and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function. In the first direction, our main positive results are the first nontrivial universal upper bounds on approximability by DNFs: (a) every Boolean function can be $\epsilon$-approximated by a DNF of size $O_\epsilon(2^n/\log n)$, and (b) every Boolean function can be $\epsilon$-approximated by a DNF of width $c_\epsilon\, n$, where $c_\epsilon < 1$. Our techniques extend broadly to give strong universal upper bounds on approximability by various depth-2 circuits that generalize DNFs, including the intersection of halfspaces, low-degree Polynomial threshold functions, and unate functions. We show that the parameters of our constructions almost match the information-theoretic inapproximability of a random function. In the second direction our mai...

In this paper, we provide normal forms and truth tables for interval-valued fuzzy logic which are analogous to those for classical logic, i.e. analogous to the disjunctive and conjunctive normal forms and truth tables for Boolean algebras. We give an algorithm for rewriting an expression to obtain its disjunctive normal form. We also give an algorithm for obtaining the disjunctive normal form of an expression from its table of truth values.

Association rule mining is a very popular data mining technique. Rules in this technique are often used to identify and represent dependencies between attributes in databases. Specifically, fuzzy association rules are rules that use the concepts of fuzzy sets and can be considered as a special case of fuzzy predicates. Many quality measures have been defined for fuzzy association rules, but all consider a specific structure: antecedent and consequence. In the case of fuzzy predicates in the normal form (i.e., conjunctive or disjunctive), it is necessary to define different quality measures that do not consider the structure as an antecedent or a consequence. The only available measure for this scenario is the fuzzy predicate truth value (FPTV), which has serious limitations. The evaluation of fuzzy predicates in the normal form through appropriate quality measures has not yet been clearly defined in the literature. Thus, we propose several quality measures specifically for fuzzy predicates in the conjunctive (CNF) and disjunctive (DNF) normal forms. Experimental studies illustrate the use of the proposed measures and allow some general conclusions about each measure.

Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Grobner ¨ basis algorithms, often using more memory than is feasible; while CNFs are solved using SAT solvers, which cannot exploit the algebra of polynomials naturally. We propose a paradigm that bridges between ANF and CNF solving techniques: the techniques are applied in an iterative manner to learn facts to augment the original problems. Experiments on over 1,100 benchmarks arising from four different applications domains demonstrate that learnt facts can significantly improve runtime and enable more benchmarks to be solved.

Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I’ll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group structure. There are also some infinite dimensional analogs but I won’t go into that here. The analytic ideas are not so different from the classical Fourier transform and Fourier inversion theories in one real variable.KeywordsUnitary RepresentationHeisenberg GroupHaar MeasureSpherical FunctionCompact Abelian GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In this paper, quantum algorithms for learning the ANF of quadratic Boolean functions are studied. We draw a conclusion about the influences of variables on quadratic functions, so that the BV algorithm can be run on them. We study the functions obtained by inversion and zero-setting of some variables in the quadratic function and show the construction of their quantum oracle. We introduce the concept of “club” to group variables that appear in quadratic terms and study the properties of clubs. Furthermore, we propose a bunch of algorithms for learning the full ANF of quadratic Boolean functions. The most efficient algorithm, among those we propose, provides an O(n) speedup over the classical one, and the number of queries is independent of the degenerate variables.

Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.