Amplification of one-way information complexity via codes and noise sensitivity

Abstract

We show a new connection between the information complexity of one-way communication problems under product distributions and a relaxed notion of list-decodable codes. As a consequence, we obtain a characterization of the information complexity of one-way problems under product distributions for any error rate based on covering numbers. This generalizes the characterization via VC dimension for constant error rates given by Kremer, Nisan, and Ron (CCC, 1999). It also provides an exponential improvement in the error rate, yielding tight bounds for a number of problems. In addition, our framework gives a new technique for analyzing the complexity of composition (e. g., XOR and OR) of one-way communication problems, connecting the difficulty of these problems to the noise sensitivity of the composing function. Using this connection, we strengthen the lower bounds obtained by Molinaro, Woodruff and Yaroslavtsev (SODA, 2013) for several problems in the distributed and streaming models, obtaining optimal lower bounds for finding the approximate closest pair of a set of points and the approximate largest entry in a matrix product. Finally, to illustrate the utility and simplicity of our framework, we show how it unifies proofs of existing 1-way lower bounds for sparse set disjointness, the indexing problem, the greater than function under product distributions, and the gap-Hamming problem under the uniform distribution.