Abstract
A function defined on the Boolean hypercube is k - Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f : F 2 n → 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 · k log 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 [2013].
Highlights
Functions defined on the Boolean hypercube {0, 1}n = Fn2 are fundamental objects in theoretical computer science
Of the 2n functions {χS}S⊆[n] defined by χS(x) = (−1) . i∈S xi This representation is known as the Fourier expansion of the function f, and the numbers f(S) are known as its Fourier coefficients
The Fourier expansion of functions plays a central role in analysis of Boolean functions and finds applications in numerous areas of theoretical computer science including learning theory, property testing, hardness of approximation, social choice theory, and cryptography
Summary
O’Donnell, Servedio, Shpilka, and Wimmer considered in [13] the problem of testing if a given Boolean function is k-Fourier-sparse or ε-far from any such function. Another problem studied there is that of deciding if a function is k-Fourier-dimensional, that is, the Fourier support, viewed as a subset of Fn2 , spans a subspace of dimension at most k, or ε-far from satisfying this property. For k-Fourier-sparsity the query complexity was a certain polynomial in k and 1/ε and√for k-Fourier-dimensionality it was O(k · 22k/ε) They proved lower bounds of Ω( k) and Ω(2k/2) respectively. These properties have recently attracted a significant amount of attention in the attempt to characterize efficient testability of them (see [24, 5] for related surveys)
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