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

@article{Tsang2013FourierSS,
title={Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture},
author={Hing Yin Tsang and Chung Hoi Wong and Ning Xie and Shengyu Zhang},
journal={2013 IEEE 54th Annual Symposium on Foundations of Computer Science},
year={2013},
pages={658-667}
}

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<sub>f</sub> for such functions is exactly the Fourier sparsity of f. Let d = deg<sub>2</sub>(f) be the F<sub>2</sub>-degree of f and DCC(f · ⊕) stand for the deterministic communication… CONTINUE READING