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We develop a novel and powerful technique for communication lower bounds, thepattern matrix method. Specifically, fix an arbitrary function f:{0,1}^n->{0,1}and let A_f be the matrix whose columns are each an application of f to somesubset of the variables x_1,x_2,...,x_{4n}. We prove that A_f has bounded-errorcommunication complexity Omega(d), where d is the approximate degree of f. Thisresult remains valid in the quantum model, regardless of prior entanglement. Inparticular, it gives a new and simple proof of Razborov's breakthrough quantumlower bounds for disjointness and other symmetric predicates. We furthercharacterize the discrepancy, approximate rank, and approximate trace norm ofA_f in terms of well-studied analytic properties of f, broadly generalizingseveral recent results on small-bias communication and agnostic learning. Themethod of this paper has recently enabled important progress in multipartycommunication complexity.

The Pattern Matrix Method (Journal Version)