On the communication exponent of distributed testing for Gaussian correlations

This work addresses distributed binary hypothesis testing, where observations at two terminals are jointly Gaussian, each one standard, with two possible correlation coefficients. We assume that one of the terminals is colocated with the decision center, and focus on a single (Stein) error exponent. Rather than the traditional exponent that is defined with respect to the source blocklength, we assume the source data to be unlimited, and consider the error exponent as a function of the communication message length. We examine two different approaches, one by quantization and the other by sending the index of the maximum, and find them to yield the same exponent. We further find that binning improves upon both approaches in the same way. Finally we compare the obtained exponents to two upper bounds and determine the optimal exponent in some very special cases.