Spectral Norm of Symmetric Functions

Abstract

The spectral norm of a Boolean function f : {0, 1} n → {−1, 1} is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as r(f ) log(n/r(f )) where r(f ) = max{r 0 , r 1 }, and r 0 and r 1 are the smallest integers less than n/2 such that f (x) or f (x) · PARITY(x) is constant for all x with x i ∈ [r 0 , n − r 1 ]. We mention some applications to the decision tree and communication complexity of symmetric functions.

Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

“…We actually do not know any counterexample for Conjecture 27 even for c = 1. Indeed, we can show that Conjecture 27 actually holds for several classes of functions, where for symmetric functions we use a result from [AFH12].…”

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“…The classic result of Nisan [33] gave a generator stretching O(log 2 n) uniformly random bits to n bits that are pseudorandom against ordered branching programs of polynomial width. 1 Despite intensive study, this is the best known seed length for ordered branching programs even of width 3, but a variety of results have shown improvements for restricted classes of ordered branching programs such as width-2 programs [39,5], and "regular" or "permutation" ordered branching programs (of constant width) [9,10,28,14,45]. 2 For width 3, hitting set generators (a relaxation of pseudorandom generators) have been constructed [42,18].…”

“…3 A recent line of work [6,24,36] has constructed pseudorandom generators for unordered, readonce, oblivious branching programs (where the bits are fed to the branching program in an arbitrary, fixed order); however, none match both the seed length and generality of Nisan's result. For unordered branching programs of length n and width w, Impagliazzo, Meka, and Zuckerman [24] give seed length O((nw) 1/2+o (1) ) improving on the linear seed length (1 − Ω(1)) · n of Bogdanov, Papakonstantinou, and Wan [6]. 4 Reingold, Steinke, and Vadhan [36] achieve seed length O(w 2 log 2 n) for the restricted class of permutation ordered branching programs, in which T i (·, b) is a permutation on [w] for all i ∈ [n] and b ∈ {0, 1}.…”

“…is a fundamental measure of complexity for Boolean functions (see, e. g., [19]), and its study has applications to learning theory [29,30], communication complexity [20,1,47,41], and circuit complexity [8,11,12]. In the study of pseudorandomness, it is well known that small-bias generators 5 with bias ε/L (which can be sampled using a seed of length O(log(n · L/ε)) [31,2]) will ε-fool any function whose Fourier mass is at most L. Width-2, read-once, oblivious branching programs have Fourier mass at most O(n) [5,39] and are thus fooled by small-bias generators with bias ε/n.…”

“…The main difference is that we need a variant of the "collision lemma" of Brody and Verbin [10]. We call a layer B i of an ordered branching program trivial if B i [0] = B i [1] and nontrivial otherwise. We say that a layer B i has a collision if there exist two edges with the same label and the same endpoint, but different start points.…”