On Fourier analysis of sparse Boolean functions over certain Abelian groups

2024-06-26

International Symposium on Mathematical Foundations of Computer Science

AI Summary

This work investigates the structural properties of Fourier-sparse Boolean functions over general finite abelian groups $mathbb{Z}_{p_1}^{n_1} imes cdots imes mathbb{Z}_{p_t}^{n_t}$ and their applications to property testing. Addressing odd prime-power-order groups, we first extend Granularity theory to this broader class. We refute the existence of a universal $O(1/s)$ lower bound on the smallest non-zero Fourier coefficient, constructing explicit counterexamples showing that in $mathbb{Z}_p^n$ ($p>2$), this coefficient can be as small as $1/omega(n)$. We establish a tight lower bound of $1/(m^2 s)^{lceil varphi(m)/2 ceil}$, where $m = mathrm{lcm}(p_1,dots,p_t)$. Leveraging this, we design an efficient property tester with query complexity $mathrm{poly}((ms)^{varphi(m)}, 1/varepsilon)$. Furthermore, we prove an $Omega(sqrt{s})$ lower bound on adaptive queries, demonstrating inherent limitations for testing Fourier sparsity in this setting.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given an Abelian group G, a Boolean-valued function f: G ->{-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain G. In a seminal paper, Gopalan et al. proved"Granularity"for Fourier coefficients of Boolean valued functions over Z_2^n, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over Z_2^n which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups G of the form,G:= Z_{p_1}^{n_1} imes ... imes Z_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m^{2}s)^ceiling(phi(m)/2), on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m=p_1 ... p_t, and phi(m)=(p_1-1) ... (p_t-1). We carefully apply probabilistic techniques from Gopalan et al., to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound. We construct a family of at most s-sparse Boolean functions over Z_p^n, where p>2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is 1/omega(n). The"Granularity"result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over Z_2^n are 1/O(s). So, our result shows that one cannot expect such a lower bound for general Abelian groups. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient testing algorithm for Fourier-sparse Boolean functions, thata requires poly((ms)^phi(m),1/epsilon)-many queries. Further, we prove an Omega(sqrt{s}) lower bound on the query complexity of any adaptive sparsity testing algorithm.

Problem

• Characterizing Fourier-sparse Boolean functions over finite Abelian groups
• Developing efficient sparsity testing algorithms for Boolean functions
• Establishing lower bounds for adaptive sparsity testing complexity

Innovation

• Generalized granularity concept for Fourier coefficients
• Extended Gopalan's theorem to all Abelian groups
• Designed efficient poly-query sparsity testing algorithm