One-way Communication and Linear Sketching for Uniform Distribution

This note is prepared based on the article titled “Linear Sketching over F2”(ECCC TR16-174) by Sampath Kannan, Elchanan Mossel and Grigory Yaroslavtsev. We quantitatively improve the parameters of Theorem 1.4 of the above work, as well as an earlier bound of ours. In particular, our result implies that for every ∈ (0, 12 ) and every constant δ > 0, the one-way communication complexity of any Boolean function f(x, y) := f(x ⊕ y) corresponding to the uniform distribution over the input domain {+1,−1} × {+1,−1} and error , is asymptotically lower bounded by the linear sketch complexity of f(x) corresponding to the uniform distribution over the input domain {+1,−1} and error (2 + δ) . Our proof is information theoretic; our improvement is obtained by studying the mutual information between Alice’s message and the evaluation of certain parities in the Fourier support of f on her input. We recall the definition of approximate Fourier dimension by Kannan et al. (TR16-174). Definition 1 (η-approximate Fourier dimension, Kannan et al. 2016). The η-approximate Fourier dimension of a Boolean function f(x) = ∑ S f̂(S)χS(x) is defined to be the smallest dimension of any linear subspace A ∈ F2 such that ∑ S∈A f̂(S) 2 ≥ η. We will need the following basic fact about the Shannon entropy of ±1valued random variables, that can be easily proved by considering the Taylor expansion of the binary entropy function H(p) about p = 12 . Fact 2. There is a universal constant k ∈ (0, 1) such that for any random variable X supported on {+1,−1}, H(X) ≤ 1− k(EX)2.