On Fourier Analysis of Sparse Boolean Functions over Certain Abelian Groups
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. [Gopalan et al., 2011] proved "Granularity" for Fourier coefficients of Boolean valued functions over ℤ₂ⁿ, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over ℤ₂ⁿ 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} × ⋯ × ℤ_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m²s)^⌈φ(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 φ(m) = (p_1-1) ⋯ (p_t-1). We carefully apply probabilistic techniques from [Gopalan et al., 2011], 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_pⁿ, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is o(1/s). 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 ℤ₂ⁿ are Ω(1/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 sparsity testing algorithm for Boolean function, which tests whether the given function is s-sparse, or ε-far from any sparse Boolean function, and it requires poly((ms)^φ(m),1/ε)-many queries. Further, we generalize the notion of degree of a Boolean function over an Abelian group 𝒢. We use it to prove an Ω(√s) lower bound on the query complexity of any adaptive sparsity testing algorithm.
Fourier coefficients
sparse
Abelian
granularity
Mathematics of computing
Theory of computation
40:1-40:16
Regular Paper
https://arxiv.org/abs/2406.18700
Sourav
Chakraborty
Sourav Chakraborty
Indian Statistical Institute Kolkata, India
https://www.isical.ac.in/~sourav/
https://orcid.org/0000-0001-9518-6204
Supported by the Science & Engineering Research Board of the DST, India, through the MATRICS grant MTR/2021/000318.
Swarnalipa
Datta
Swarnalipa Datta
Indian Statistical Institute Kolkata, India
Pranjal
Dutta
Pranjal Dutta
National University of Singapore, Singapore
https://sites.google.com/view/pduttashomepage/home
https://orcid.org/0000-0001-9137-9025
Supported by the project "Foundation of Lattice-based Cryptography", funded by NUS-NCS Joint Laboratory for Cyber Security, Singapore.
Arijit
Ghosh
Arijit Ghosh
Indian Statistical Institute Kolkata, India
https://www.isical.ac.in/arijit-ghosh
https://orcid.org/0000-0002-2821-265X
Arijit Ghosh is partially supported by the Science & Engineering Research Board of the DST, India, through the MATRICS grant MTR/2023/001527.
Swagato
Sanyal
Swagato Sanyal
Indian Institute of Technology Kharagpur, India
https://cse.iitkgp.ac.in/~swagato/
https://orcid.org/0000-0003-1546-7749
10.4230/LIPIcs.MFCS.2024.40
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