The Pattern Matrix Method

We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f: \0,1\ⁿ →\0,1\, and let Af be the matrix whose columns are each an application of f to some subset of the variables x₁,x₂,…,x₄ₙ. We prove that Af has bounded-error communication complexity Ω(d), where d is the approximate degree of $f.$ This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of Af in terms of well-studied analytic properties of $f,$ broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has also enabled important progress in multiparty communication complexity.