The Pattern Matrix Method (Journal Version)

TL;DR

The paper introduces the pattern matrix method, a new technique for establishing communication lower bounds that connects the complexity of a function to the properties of a constructed matrix, with applications in quantum and classical models.

Contribution

It presents the pattern matrix method, linking function complexity to matrix properties, and applies it to quantum and classical communication complexity, generalizing previous results.

Findings

Proves Omega(d) lower bounds for communication complexity based on approximate degree.

Characterizes discrepancy, approximate rank, and trace norm via analytic properties of functions.

Provides a simple proof of quantum lower bounds for disjointness and symmetric predicates.

Abstract

We develop a novel and powerful technique for communication lower bounds, the pattern 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 some subset of the variables x_1,x_2,...,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 recently enabled important progress in multiparty communication complexity.