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Implicit Sensing for Fourier Sparse Boolean Functions

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
Learning TheoryFourier AnalysisFourier SparsitySublinear AlgorithmProperty TestingComprssed Sensing
Abstract:

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:F2n→{−1,+1}f : \mathbb{F}_2^n \to \{-1, +1\}, decide whether it is ss-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 O~(s4)\widetilde{O}(s^4), independent of the ambient dimension. On the lower bound side, we prove that any tester requires at least Ω(s)\Omega(s) queries. Both bounds improve upon the best known results of Gopalan et al.\ (SICOMP 2011), who presented a tester with query complexity O~(s14)\widetilde{O}(s^{14}) and a lower bound of Ω(s)\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 ℓ1\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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper addresses Fourier sparsity testing for Boolean functions, developing algorithms to decide whether a function is s-Fourier-sparse or far from any such function. It resides in the 'Direct Fourier Sparsity Testing' leaf of the taxonomy, which contains only three papers total, including this work. This leaf sits within the broader 'Fourier Sparsity Testing and Property Testing' branch, indicating a relatively focused research direction. The small number of sibling papers suggests this is a specialized but not overcrowded area, with room for algorithmic improvements and new techniques.

The taxonomy reveals neighboring research directions including 'Related Property Testing' (testing Fourier dimension and concentrated functions) and 'Classical Learning via Fourier Analysis' (reconstruction algorithms for sparse functions). The original paper's position bridges pure property testing and learning theory: while it focuses on verification rather than reconstruction, its techniques may inform learning algorithms that assume prior knowledge of sparsity. The taxonomy's scope notes clarify that this work excludes learning algorithms and general property testing frameworks, positioning it as a foundational testing problem rather than an application-driven approach.

Among the three contributions analyzed, the literature search examined 17 candidates total. The improved upper bound algorithm (Contribution 1) examined 10 candidates with no clear refutations, suggesting relative novelty in this direction. However, the improved lower bound (Contribution 2) and novel sampling method for parity decision trees (Contribution 3) each found at least one refutable candidate among 6 and 1 examined respectively. This indicates that while the upper bound algorithm may represent a fresh approach, the lower bound and sampling techniques have more substantial prior work within the limited search scope.

Based on the top-17 semantic matches examined, the paper appears to make incremental progress on a well-defined problem with limited prior work. The small taxonomy leaf size and focused search scope mean this assessment captures nearby work but may not reflect the full landscape of related techniques in broader property testing or Fourier analysis. The mixed refutation results suggest some contributions build more directly on existing foundations than others.

Taxonomy

Core-task Taxonomy Papers
34
3
Claimed Contributions
17
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: Testing Fourier sparsity of Boolean functions. The field centers on determining whether a Boolean function has a sparse Fourier representation, meaning only a small number of its Fourier coefficients are nonzero. The taxonomy reveals several interconnected branches: direct testing algorithms that query functions to verify sparsity (e.g., Fast Fourier Sparsity Testing[6], Testing Fourier Dimensionality[8]); learning approaches that reconstruct or approximate sparse functions (e.g., Learning Decision Trees Fourier[21], Quantum Learning Concentrated[18]); complexity-theoretic investigations of structural bounds and measures like Fourier dimension (e.g., Fourier Sparsity Dimension[5], Sublinear Fourier Dimension Bounds[17]); representation-theoretic studies extending beyond Boolean domains (e.g., Fourier Analysis Abelian Groups[12], Sparse Legendre Expansions[22]); and application-driven work in areas such as cryptography and communication complexity (e.g., Boolean Analysis Communication Complexity[29]). These branches collectively address how sparsity interacts with computational efficiency, learnability, and structural properties of functions. Recent activity highlights trade-offs between query complexity and sparsity guarantees, with some works exploring sublinear-time testing under various noise models and others examining the interplay between sparsity and other complexity measures like decision tree depth (Parity Decision Trees[1], Decision Tree Rank[19]). The original paper, Implicit Sensing Fourier Sparse[0], sits within the direct testing branch alongside Fast Fourier Sparsity Testing[6] and Price of Parsimony[33]. While Fast Fourier Sparsity Testing[6] emphasizes explicit query-efficient algorithms, Implicit Sensing Fourier Sparse[0] appears to focus on implicit or adaptive sensing strategies that may reduce the number of queries needed by leveraging structural assumptions. This contrasts with learning-oriented approaches like Active Learning Walsh Hadamard[34], which prioritize reconstruction over mere verification, illustrating the spectrum from pure property testing to full function recovery.

Claimed Contributions

Improved upper bound algorithm for Fourier sparsity testing

The authors present a non-adaptive algorithm for testing whether a Boolean function is s-Fourier sparse or far from any such function, achieving query complexity eO(s4), which improves upon the previous best bound of eO(s14). The algorithm combines a refined sampler from junta testing with l1-minimization-based compressed sensing techniques.

10 retrieved papers
Improved lower bound for Fourier sparsity testing

The authors establish a new lower bound of Ω(s) queries for any adaptive or non-adaptive tester for Fourier sparsity, improving quadratically over the previous Ω(√s) bound. This is achieved via a reduction from communication complexity using structural properties of Maiorana–McFarland functions.

6 retrieved papers
Can Refute
Novel sampling method for parity decision trees

The authors introduce a new technique for generating uniform coset samples from the Fourier span of Boolean functions, which serves as a key component in their upper bound algorithm. This method refines the sampler notion from the junta testing framework and enables query-efficient implicit learning of Fourier-sparse functions.

1 retrieved paper
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Improved upper bound algorithm for Fourier sparsity testing

The authors present a non-adaptive algorithm for testing whether a Boolean function is s-Fourier sparse or far from any such function, achieving query complexity eO(s4), which improves upon the previous best bound of eO(s14). The algorithm combines a refined sampler from junta testing with l1-minimization-based compressed sensing techniques.

Contribution

Improved lower bound for Fourier sparsity testing

The authors establish a new lower bound of Ω(s) queries for any adaptive or non-adaptive tester for Fourier sparsity, improving quadratically over the previous Ω(√s) bound. This is achieved via a reduction from communication complexity using structural properties of Maiorana–McFarland functions.

Contribution

Novel sampling method for parity decision trees

The authors introduce a new technique for generating uniform coset samples from the Fourier span of Boolean functions, which serves as a key component in their upper bound algorithm. This method refines the sampler notion from the junta testing framework and enables query-efficient implicit learning of Fourier-sparse functions.

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