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

Abstract

We study Boolean functions with sparse Fourier spectrum or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f (x ⊕ y) -a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communication matrix M f for such functions is exactly the Fourier sparsity of f . Let d = deg 2 (f ) be the F2-degree of f and D CC (f • ⊕) stand for the deterministic communication complexity for f (x ⊕ y). We show thatThis improves the (trivial) linear bound by nearly a quadratic factor. We obtain our results through a degree-reduction protocol based on a variant of polynomial rank, and actually conjecture that the communication cost of our protocol is at most log O(1) rank(M f ). The above bounds are obtained from different analysis for the number of parity queries required to reduce f 's F2-degree. Our bounds also hold for the parity decision tree complexity of f , a measure that is no less than the communication complexity.Along the way we also prove several structural results about Boolean functions with small Fourier sparsity f 0 or spectral norm f 1, which could be of independent interest. For functions f with constant F2-degree, we show that: 1) f can be written as the summation of quasi-polynomially many indicator functions of subspaces with ±-signs, improving the previous doubly exponential upper bound by Green and Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having a small parity decision tree complexity; and 3) f depends only on polylog f 1 linear functions of input variables. For functions f with small spectral norm, we show that: 1) there is an affine subspace of co-dimension O( f 1) on which f (x) is a constant, and 2) there is a parity decision tree of depth O( f 1 log f 0) for computing f .

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“…After almost a decade of efforts, the conjecture has been established for several classes of XOR-function, such as symmetric functions [20], monotone functions and linear threshold functions [12], constant F 2 -degree functions [17]. A different line of work close to ours is the simulation theorem in [13,20,15,3,4].…”

“…We adapt a protocol introduced by Tsang et al [17]. The main step is to exhibit a large monochromatic affine subspace for f if the communication complexity of F is small.…”

“…This conjecture, proposed by Lovász and Saks in [8], asserts that the communication complexity of F and log rank (M F ) are polynomially equivalent for any 2-argument total boolean function F , where M F = [F (x, y)] x,y is the communication matrix of F . Readers may refer to [17] for more discussion on the conjecture. The conjecture is notoriously hard to attack.…”

“…The communication complexity of XOR functions has been studied extensively in the last decade [20,6,12,7,17,19]. A nice feature of XOR-functions is that the rank of the communication matrix M F is exactly the Fourier sparsity of f .…”

“…Using such an approach, the Log-Rank Conjecture has been established for several subclasses of XOR-functions [12,17].…”