A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most knonzero Fourier coefficients. For a function f: Fn2 → ℝ and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n·klog k). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz .
- Fourier-sparse Boolean functions
- Learning theory
- Property testing
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics