The CEO Problem with rth Power of Difference and Logarithmic Distortions
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The CEO problem has received a lot of attention since Berger et al. first investigated it, however, there are limited results on non-Gaussian models with non-quadratic distortion measures. In this work, we extend the CEO problem to two continuous alphabet settings with general rth power of difference and logarithmic distortions, and study asymptotics of distortion as the number of agents and sum rate grow without bound. The first setting is a regular source-observation model, such as jointly Gaussian, with difference distortion and we show that the distortion decays at R_sum up to a multiplicative constant. We use sample median estimation following the Berger-Tung scheme for achievability and the Shannon lower bound for the converse. The other setting is a non-regular source-observation model, such as copula or uniform additive noise models, with difference distortion for which estimation-theoretic regularity conditions do not hold. The optimal decay R_sum is obtained for the non-regular model by midrange estimator following the Berger-Tung scheme for achievability and the Chazan-Ziv-Zakai bound for the converse. Lastly, we provide a condition for the regular model, under which quadratic and logarithmic distortions are asymptotically equivalent by entropy power relation as the number of agents grows. This proof relies on the Bernstein-von Mises theorem.
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