Testing Fourier Dimensionality and Sparsity

Abstract

Abstract. 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 first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of F n 2 (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm -even an adaptive onemust have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an "implicit learning" algorithm that lets us test any sub-property of Fourier concision. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [13].

Testing Linear-Invariant Function Isomorphism

“…It is shown that r-sparsity and k-dimensionality are constantquery testable in [13]. As a corollary of Theorem 4, these properties are tolerantly constant-query testable.…”

“…Proof sketch of Theorem 1.1 We prove our main theorem in a similar manner to the implicit learning method used in [14] and [13]. Similarly to [13], our algorithm finds all the large Fourier coefficients of the unknown function f using an implicit version of the Goldreich-Levin algorithm [15]; we call our algorithm Implicit Sieve.…”

“…Similarly to [13], our algorithm finds all the large Fourier coefficients of the unknown function f using an implicit version of the Goldreich-Levin algorithm [15]; we call our algorithm Implicit Sieve. Given a set of vectors M, it outputs…”

“…The approach in [13] works when f is k-dimensional; our algorithm Implicit Sieve gives correct output with no assumptions about f .…”

“…Once Implicit Sieve is run, rather than checking consistency as in [14] and [13], our algorithm produces a sparse polynomial that approximates f . This can be done by estimating f (α i ) from the output of Implicit Sieve.…”