Testing fourier dimensionality and sparsity

Parikshit Gopalan*, Ryan O'Donnell, Rocco A. Servedio, Amir Shpilka, Karl Wimmer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of F2n. In both cases we also prove lower bounds showing that any testing algorithm-even an adaptive one-must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Building on these results, we give an "implicit learning" algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].

Original languageEnglish
Pages (from-to)1075-1100
Number of pages26
JournalSIAM Journal on Computing
Issue number4
StatePublished - 2011
Externally publishedYes


  • Discrete fourier analysis
  • Fourier spectrum
  • Local testability
  • Property testing


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