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.
Many of the recent advances in the field of communication complexity @cite_21 have followed from using various notions of information complexity. Earlier notions of information complexity appeared implicitly in several works @cite_25 @cite_22 @cite_13 , and was first explicitly defined in @cite_26 and further developed in @cite_16 . Information complexity has been extensively used or studied in the recent communication complexity literature (e.g., @cite_27 @cite_14 @cite_3 @cite_20 @cite_7 @cite_11 @cite_10 @cite_0 @cite_6 ). The notion was also adapted to specialized models or tasks @cite_12 @cite_1 @cite_8 @cite_19 .
@cite_22 @cite_3 @cite_10 @cite_20 @cite_8 @cite_21 @cite_26 @cite_7 @cite_6 @cite_19 @cite_27 @cite_16 @cite_25 @cite_12 @cite_14 @cite_1 @cite_0 @cite_13 @cite_11
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Three decades of research in communication complexity have led to the invention of a number of techniques to lower bound randomized communication complexity. The majority of these techniques involve properties of large submatrices (rectangles) of the truth-table matrix defining a communication problem. The only technique that does not quite fit is information complexity, which has been investigated over the last decade. Here, we connect information complexity to one of the most powerful "rectangular" techniques: the recently-introduced smooth corruption (or "smooth rectangle") bound. We show that the former subsumes the latter under rectangular input distributions. We conjecture that this subsumption holds more generally, under arbitrary distributions, which would resolve the long-standing direct sum question for randomized communication. As an application, we obtain an optimal @math lower bound on the information complexity---under the uniform distribution ---of the so-called orthogonality problem (ORT), which is in turn closely related to the much-studied Gap-Hamming-Distance (GHD). The proof of this bound is along the lines of recent communication lower bounds for GHD, but we encounter a surprising amount of additional technical detail. We show an exponential gap between communication complexity and external information complexity, by analyzing a communication task suggested as a candidate by Braverman. Previously, only a separation of communication complexity and internal information complexity was known. More precisely, we obtain an explicit example of a search problem with external information complexity ≤ O(k), with respect to any input distribution, and distributional communication complexity ≥ 2k, with respect to some input distribution. In particular, this shows that a communication protocol cannot always be compressed to its external information. By a result of Braverman, our gap is the largest possible. Moreover, since the upper bound of O(k) on the external information complexity of the problem is obtained with respect to any input distribution, our result implies an exponential gap between communication complexity and information complexity (both internal and external) in the non-distributional setting of Braverman. In this setting, no gap was previously known, even for internal information complexity. We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥ 2k. This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [1], our gap is the largest possible. By a result of Braverman and Rao [2], 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. The primary goal of this paper is to define and study the interactive information complexity of functions. Let f(x,y) be a function, and suppose Alice is given x and Bob is given y. Informally, the interactive information complexity IC(f) of f is the least amount of information Alice and Bob need to reveal to each other to compute f. Previously, information complexity has been defined with respect to a prior distribution on the input pairs (x,y). Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity IC(f). In particular, we show that IC(f) is equal to the amortized (randomized) communication complexity of f. We also show a direct sum theorem for IC(f) and give the first general connection between information complexity and (non-amortized) communication complexity. This connection implies that a non-trivial exchange of information is required when solving problems that have non-trivial communication complexity. We explore the information complexity of two specific problems - Equality and Disjointness. We show that only a constant amount of information needs to be exchanged when solving Equality with no errors, while solving Disjointness with a constant error probability requires the parties to reveal a linear amount of information to each other.