Boolean Cube Research Articles

Central limit theorem of overlap for the mean field Ghatak–Sherrington model

The Ghatak–Sherrington spin glass model is a random probability measure defined on the configuration space {0,±1,±2,…,±S}N with system size N and S⩾1 finite. This generalizes the classical Sherrington–Kirkpatrick (SK) model on the boolean cube {−1, +1}N to capture more complex behaviors, including the spontaneous inverse freezing phenomenon. We give a quantitative joint central limit theorem for the overlap and self-overlap array at sufficiently high temperature under arbitrary crystal and external fields. Our proof uses the moment method combined with the cavity approach. Compared to the SK model, the main challenge comes from the non-trivial self-overlap terms that correlate with the standard overlap terms.

Asymptotic Insights for Projection, Gordon–Lewis, and Sidon Constants in Boolean Cube Function Spaces

Abstract The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^{N}$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^{N}$ that have Fourier–Walsh expansions supported on a fixed family $\mathcal{S}$ of subsets of $\{1, \ldots , N\}$. Our investigation centers on the projection, Sidon, and Gordon–Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.

Sketching Approximability of All Finite CSPs

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Linear hashing in the Boolean cube with clusters of limited size

Показано, что для произвольного подмножества $n$-мерного булева куба из $M$ элементов существует линейная хеш-функция, кластеры которой состоят не более чем из $a$ элементов, и ранг которой не превосходит ${2\log_2 M-2\log_2 a+\mathcal O(1)}$.

Irreducible Subcube Partitions

A subcube partition is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates.
 We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size.
 Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.

Noncommutative Bohnenblust–Hille inequalities

Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes.

Octopuses in the Boolean cube: Families with pairwise small intersections, part I

Let F1,…,Fℓ be families of subsets of {1,…,n}. Suppose that for distinct k,k′ and arbitrary F1∈Fk,F2∈Fk′ we have |F1∩F2|⩽m. What is the maximal value of |F1|…|Fℓ|? In this work we find the asymptotic of this product as n tends to infinity for constant ℓ and m.This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number of facets and vertices in a two-level polytope. This conjecture was recently resolved by Weltge and the first author.The main result can be rephrased in terms of colorings. We give an asymptotic answer to the following question. Given an edge coloring of a complete m-uniform hypergraph into ℓ colors, what is the maximum of ∏Mi, where Mi is the number of monochromatic cliques in i-th color?

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

On the Padovan Codes and the Padovan Cubes

We present a new interconnection topology called the Padovan cube. Despite their asymmetric and relatively sparse interconnections, the Padovan cubes are shown to possess attractive recurrent structures. Since they can be embedded in a subgraph of the Boolean cube and can have a Fibonacci cube as a subgraph, and since they are also a supergraph of other structures, it is possible that the Padovan cubes can be useful in fault-tolerant computing. For a graph with n vertices, we characterize the Padovan cubes. We also include the number of edges, decompositions, and embeddings, as well as the diameter of the Padovan cubes.

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Categorical data plays an important part in machine learning research and appears in a variety of applications. Models that can express large classes of real-valued functions on the Boolean cube are useful for problems involving discrete-valued data types, including those which are not Boolean. To this date, the commonly used schemes for embedding classical data into variational quantum machine learning models encode continuous values. Here we investigate quantum embeddings for encoding Boolean-valued data into parameterized quantum circuits used for machine learning tasks. We narrow down representability conditions for functions on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional Boolean cube with respect to previously known results, using two quantum embeddings: a phase embedding and an embedding based on quantum random access codes. We show that for any real-valued function on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional Boolean cube, there exists a variational linear quantum model based on a phase embedding using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> qubits that can represent it and an ensemble of such models using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d \lt n$</tex-math></inline-formula> qubits that can express any function with degree at most d. Additionally, we prove that variational linear quantum models that use the quantum random access code embedding can express functions on the Boolean cube with degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d \leq \lceil \frac{n}{3}\rceil$</tex-math></inline-formula> using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lceil \frac{n}{3}\rceil$</tex-math></inline-formula> qubits, and that an ensemble of such models can represent any function on the Boolean cube with degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d \leq \lceil \frac{n}{3}\rceil$</tex-math></inline-formula> . Furthermore, we discuss the potential benefits of each embedding and the impact of serial repetitions. Lastly, we demonstrate the use of the embeddings presented by performing numerical simulations and experiments on IBM quantum processors using the Qiskit machine learning framework.

Nonoverlapping Convex Polytopes with Vertices in a Boolean Cube and Other Problems in Coding Theory

Nonoverlapping Convex Polytopes with Vertices in a Boolean Cube and Other Problems in Coding Theory

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber's-Type Inequalities for the Second Rényi Entropy.

Let , , be the noise operator acting on functions on the boolean cube . Let f be a distribution on and let . We prove tight Mrs. Gerber-type results for the second Rényi entropy of which take into account the value of the Rényi entropy of f. For a general function f on we prove tight hypercontractive inequalities for the norm of which take into account the ratio between and norms of f.

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Sufficiently Informative and Relevant Features: An Information-Theoretic and Fourier-Based Characterization

A fundamental challenge in learning is the presence of nonlinear redundancies and dependencies in the data. To address this, we propose a Fourier-based approach to characterize feature redundancies, in unsupervised learning, and feature-label dependencies, in the supervised variant of the problem. We first develop a novel Fourier expansion for functions (more generally stochastic mappings) of correlated binary random variables. This is a generalization of the standard Fourier expansion on the Boolean cube beyond product probability spaces. As an important application of this analysis, we investigate learning with feature subset selection. In the unsupervised variant of this problem, we characterize feature redundancies via the Shannon entropy and group the features into sufficiently informative and redundant. Then, we make a connection to the proposed Fourier expansion and derive an upper bound on the joint entropy. Based on that, we propose a measure to quantify feature redundancies and present an unsupervised learning algorithm. We test our method on various real-world and synthetic datasets and demonstrate improvements on conventional unsupervised feature selection techniques. Then, we investigate the supervised feature subset selection and reformulate it in the Fourier domain. Bridging the Bayesian error rate with the Fourier coefficients, we demonstrate that the Fourier expansion provides a powerful tool to characterize nonlinear feature-label dependencies. Further, we introduce a computationally efficient measure for selecting relevant features. Via a theoretical analysis, we show that our proposed measure finds provably <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">asymptotically optimal</i> feature subsets. Lastly, we present an algorithm based on this measure and via numerical experiments demonstrate its improvements on various supervised feature selection algorithms.

Private Sampling: A Noiseless Approach for Generating Differentially Private Synthetic Data

In a world where artificial intelligence and data science become omnipresent, data sharing is increasingly locking horns with data-privacy concerns. Differential privacy has emerged as a rigorous framework for protecting individual privacy in a statistical database, while releasing useful statistical information about the database. The standard way to implement differential privacy is to inject a sufficient amount of noise into the data. However, in addition to other limitations of differential privacy, this process of adding noise will affect data accuracy and utility. Another approach to enable privacy in data sharing is based on the concept of synthetic data. The goal of synthetic data is to create an as-realistic-as-possible dataset, one that not only maintains the nuances of the original data, but does so without risk of exposing sensitive information. The combination of differential privacy with synthetic data has been suggested as a best-of-both-worlds solutions. In this work, we propose the first noisefree method to construct differentially synthetic data; we do this through a mechanism called private sampling. Using the Boolean cube as benchmark data model, we derive explicit bounds on accuracy and privacy of the constructed synthetic data. The key mathematical tools are hypercontractivity, duality, and empirical processes. A core ingredient of our sampling mechanism is a rigorous marginal correction method, which has the remarkable property that importance reweighting can be utilized to exactly match the marginals of the sample to the marginals of the population.

On the Extension Complexity of Polytopes Separating Subsets of the Boolean Cube

We show that for every \(A\subseteq \{0,1\}^n\), there exists a polytope \(P\subseteq \mathbb {R}^n\) with \(P \cap \{0,1\}^n = A\) and extension complexity \(O(2^{n/2})\), and that there exists an \(A\subseteq \{0,1\}^n\) such that the extension complexity of any P with \(P\cap \{0,1\}^n = A\) must be at least \(2^{{n}(1-o(1))/3}\). We also remark that the extension complexity of any 0/1-polytope in \(\mathbb {R}^n\) is at most \(O(2^n/n)\) and pose the problem whether the upper bound can be improved to \(O(2^{cn})\), for \(c<1\).

Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes

Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least n hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for polynomial covers of weight-determined sets of strictly unimodal uniform (SU2) grids.&#x0D; As a central tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) closures of symmetric sets of the Boolean cube. In fact, we obtain a characterization that concerns, more generally, weight-determined sets of SU2 grids. However, in this generality, our characterization is not of the Z-closures but of a new variant of Z-closures defined exclusively for weight-determined sets, which coincides with the Z-closures in the Boolean cube setting, for symmetric sets. This characterization admits a linear time algorithm, and may also be of independent interest. Indeed, as further applications, we (i) give an alternate proof of a lemma by Alon et al. (IEEE Trans. Inform. Theory, 1988), and (ii) characterize the certifying degrees of weight-determined sets.&#x0D; In the Boolean cube setting, our above characterization can also be derived using a result of Bernasconi and Egidi (Inf. Comput., 1999) that determines the affine Hilbert functions of symmetric sets. However, our proof is independent of this result, works for all SU2 grids, and could be regarded as being more combinatorial.&#x0D; We also introduce another new variant of Z-closures to understand better the difference between the hyperplane and polynomial covering problems over uniform grids. Finally, we conclude by introducing a third variant of our covering problems and conjecturing its solution in the Boolean cube setting.

Weights of exact threshold functions

We consider Boolean exact threshold functions defined by linear equations and, more generally, polynomials of degree . We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of the integer coefficients of linear equations required to express every possible intersection of a hyperplane in and the Boolean cube or, in the general case, intersections of hypersurfaces of degree in and . In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension of the affine subspace spanned by the solutions and give upper and lower bounds in this case as well. There is a substantial gap between these bounds, a challenge for future work.