We introduce a new information-theoretic complexity measure @math for 2-party functions which is a lower-bound on communication complexity, and has the two leading lower-bounds on communication complexity as its natural relaxations: (external) information complexity ( @math ) and logarithm of partition complexity ( @math ), which have so far appeared conceptually quite different from each other. @math is an external information complexity measure based on R 'enyi mutual information of order infinity. In the definition of @math , relaxing the order of R 'enyi mutual information from infinity to 1 yields @math , while @math is obtained by replacing protocol transcripts with what we term "pseudotranscripts," which omits the interactive nature of a protocol, but only requires that the probability of any transcript given the inputs @math and @math to the two parties, factorizes into two terms which depend on @math and @math separately. Further understanding @math might have consequences for important direct-sum problems in communication complexity, as it lies between communication complexity and information complexity. We also show that applying both the above relaxations simultaneously to @math gives a complexity measure that is lower-bounded by the (log of) relaxed partition complexity, a complexity measure introduced by (FOCS 2012). We obtain a sharper connection between (external) information complexity and relaxed partition complexity than , using an arguably more direct proof.
The main contribution of this work is to relate information complexity to "tension" [Prabhakaran and Prabhakaran, 2014] - an information-theoretic quantity defined with no reference to protocols - and to illustrate that it allows deriving strong lower-bounds on information complexity. In particular, we use a very special case of this connection to give a quantitatively tighter connection between information complexity and discrepancy than the one in the work of Braverman and Weinstein (2012) (albeit, restricted to independent inputs). Further, as tension is in fact a multi-dimensional notion, it enables us to bound the 2-dimensional region that represents the trade-off between the amounts of communication in the two directions in a 2-party protocol. This work is also intended to highlight tension as a fundamental measure of correlation between a pair of random variables, with rich connections to a variety of questions in computer science and information theory.