Abstract
Recently, Forster [7] proved a new lower bound on probabilistic communication complexity in terms of the operator norm of the communication matrix. In this paper, we want to exploit the various relations between communication complexity of distributed Boolean functions, geometric questions related to half space representations of these functions, and the computational complexity of these functions in various restricted models of computation. In order to widen the range of applicability of Forster’s bound, we start with the derivation of a generalized lower bound. We present a concrete family of distributed Boolean functions where the generalized bound leads to a linear lower bound on the probabilistic communication complexity (and thus to an exponential lower bound on the number of Euclidean dimensions needed for a successful half space representation), whereas the old bound fails. We move on to a geometric characterization of the well known communication complexity class C-PP in terms of half space representations achieving a large margin. Our characterization hints to a close connection between the bounded error model of probabilistic communication complexity and the area of large margin classification. In the final section of the paper, we describe how our techniques can be used to prove exponential lower bounds on the size of depth-2 threshold circuits (with still some technical restrictions). Similar results can be obtained for read-k-times randomized ordered binary decision diagram and related models.
This work has been supported in part by the ESPRIT Working Group in Neural and Computational Learning II, NeuroCOLT2, No. 27150. The first, fifth, and sixth author was supported by the Deutsche Forschungsgemeinschaft Grant SI498/4-1. The third author was partially supported by NSF Grants CCR-9988359 and DMS- 9729992.
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Forster, J., Krause, M., Lokam, S.V., Mubarakzjanov, R., Schmitt, N., Simon, H.U. (2001). Relations Between Communication Complexity, Linear Arrangements, and Computational Complexity. In: Hariharan, R., Vinay, V., Mukund, M. (eds) FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2001. Lecture Notes in Computer Science, vol 2245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45294-X_15
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DOI: https://doi.org/10.1007/3-540-45294-X_15
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