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The main mathematical result of this paper may be stated as follows: Given a matrix M ∈ {-1,1}n × n and any matrix M ∈ Rn × n such that sign(Mij)=Mij for all i,j, then rank(M)≥n/||M||. Here ||M|| denotes the spectral norm of the matrix M.This implies a general lower bound on the complexity of unbounded error probabilistic communication protocols. As a simple consequence, we obtain the first linear lower bound on the complexity of unbounded error probabilistic communication protocols for the functions defined by Hadamard matrices. This solves a long-standing open problem stated by Paturi and Simon (J. Comput. System Sci. 33 (1986) 106).We also give an upper bound on the margin of any embedding of a concept class in half spaces. Such bounds are of interest to problems in learning theory.