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

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 = deg₂(f) be the F₂-degree of f and DCC(f · ⊕) stand for the deterministic communication complexity for f(x ⊕ y). We show that 1) DCC(f · ⊕) = O(2^d2/2 log^d-2 ∥f̂∥₁). In particular, the Log-rank conjecture holds for XOR functions with constant F₂-degree. 2) D^CC(f · ⊕) = O(d∥f̂∥₁) = O(√(rank(M_f))). This 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 ∥f̂∥₀ or spectral norm ∥f̂∥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∥f̂∥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 ∥f̂∥₁ on which f(x) is a constant, and 2) there is a parity decision ∥f̂∥₁log∥f̂∥₀ for computing f.