We develop a new method to prove communication lower bounds for composed functions of the form f • g n , where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f •g n can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only "query" a few inputs of f as encoded by the

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... g. Consequently, we characterize the communication complexity of f • g n in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of f . These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc . These results answer questions of Klauck [We also show that the approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. [Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), Springer, Berlin, 2014, pp. 701-712] for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the clique versus independent set problem.