The Pattern Matrix Method

We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function $f: \{0,1\}^n \to\{0,1\}$, and let $A_f$ be the matrix whose columns are each an application of $f$ to some subset of the variables $x_1,x_2,\ldots,x_{4n}.$ We prove that $A_f$ has bounded-error communication complexity $\Omega(d),$ 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 $A_f$ 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.