Implicit Sensing for Fourier Sparse Boolean Functions

Boolean functions constitute a fundamental object of study in machine learning and, more broadly, in theoretical computer science. Among their various complexity measures, Fourier sparsity, defined as the number of nonzero Fourier coefficients in a Boolean function’s Fourier expansion, serves as a key indicator of structural simplicity. For over three decades, learning Boolean functions with sparse Fourier representations has been a central focus of computational learning theory. A notable achievement in this line of work is the development of learning algorithms whose complexity primarily depends on the Fourier sparsity parameter. However, these approaches generally assume prior knowledge of this parameter. In this work, we address this gap in the literature on learning Fourier-sparse Boolean functions. Specifically, we study the problem of Fourier sparsity testing: given query access to a Boolean function $f : \mathbb{F}_2^n \to \{-1, +1\}$, decide whether it is $s$-Fourier sparse or far (under Hamming distance) from every such function. Our contributions are twofold. On the algorithmic side, we design a new tester with query complexity $\widetilde{O}(s^4)$, independent of the ambient dimension. On the lower bound side, we prove that any tester requires at least $\Omega(s)$ queries. Both bounds improve upon the best known results of Gopalan et al.\ (SICOMP 2011), who presented a tester with query complexity $\widetilde{O}(s^{14})$ and a lower bound of $\Omega(\sqrt{s})$. For our upper bound, we introduce a refined notion of a sampler from the junta testing framework of Chakraborty et al.\ (ICALP 2011) and combine it with $\ell_1$-minimization-based compressed sensing techniques to construct our tester. In the process, we develop a novel method for sampling the leaves of parity decision trees associated with Fourier-sparse Boolean functions. The lower bound is obtained via a reduction from communication complexity, crucially leveraging structural properties of the Fourier coefficients of a specific class of cryptographically hard functions.