Exponential separation of information and communication for boolean functions

We show an exponential gap between communication complexity and information complexity by giving an explicit example of a partial boolean function with information complexity ≤O(k), and distributional communication complexity ≥2^k. This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [2015], our gap is the largest possible. By a result of Braverman and Rao [2014], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold, answering a long-standing open problem. Another (conceptual) contribution of our work is the relative discrepancy method, a new rectangle-based method for proving communication complexity lower bounds for boolean functions, powerful enough to separate information complexity and communication complexity.