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This paper extends the authors' prior work on "E-type" (Exponential family type) channels. The channels considered here have transition kernels induced by an exponential family with a two-component sufficient statistic composed of an input-output distortion function and an output cost function. They demonstrate the existence of a mutual information saddle point in any E-type channel for which there exists a source distribution such that the induced output distribution is maximum-entropy under an output cost constraint. For additive-noise E-type channels, they provide necessary and sufficient conditions on the existence of saddle points which coincide with convolution divisibility of the additive noise law.
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