Spectral analysis of Boolean functions as a graph eigenvalue problem

Testing Fourier Dimensionality and Sparsity

“…The first inequality above is not difficult (see e.g. [8,Lemma 3]) and the second one is essentially immediate. Either of the above inequalities can be quite loose; for the first inequality, the inner product function on n variables has deg 2 (f ) = 2 but log sp(f ) = n. For the second inequality, the addressing function with 1 2 log s addressing variables and s 1/2 addressee variables can be shown to be s-sparse but has dim(f ) ≥ s 1/2 .…”

On the Walsh-Fourier analysis of Boolean functions

“…For f ∈ F, we consider the set f −1 (1) and the following graph G f where the vertex set is F n 2 , and the edge set is defined by For a detailed study on this topic, see [1] and [2]. We denote Sym(E) the group of permutations on the set E, and for each σ ∈ Sym(E), ε(σ) the parity +1 or −1 of σ.…”

Graph eigenvalues and Walsh spectrum of Boolean functions

“…In fact, they are more related than one might initially guess, as we shall see next. The attempt in the present paper (and in a few other works, see [3,4,5]) is to push further the connection between two very intriguing topics, bent functions and strongly regular graphs, with the hope that the investigation will shed more light into the constructions of both. We would like to invite researchers in these two areas to collaborate for the benefit of all parties.…”