Lower bounds for elimination via weak regularity

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. [1] and later studied by Beimel et al. [4] for its connection to the famous direct sum question. In this problem, let f: {0, 1}2n → {0,1} be any boolean function. Alice and Bob get k inputs x1,⋯, xk and y1,⋯, yk respectively, with xi, yi ∈ {0, 1}n. They want to output a k-bit vector v, such that there exists one index i for which vi = f(xi,yi). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson [19]. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola [23], to show that Greater-Than is weakly regular.