The rate-information trade-off for Gaussian vector channels

We consider lossy compression of the output of a Gaussian vector channel. This is relevant for quantizer design in rate-limited feedback and in receiver front-ends. In particular, we study the trade-off between compression rate and mutual information between channel input and compressed channel output. Using the Gaussian information bottleneck we provide closed-form expressions for the information-rate function and the rate-information function. We prove that optimal compression in the rate distortion sense with squared-error distortion does not achieve the optimal rate-information trade-off. The suboptimality of the rate distortion approach is quantified and we give an upper bound on the gap to the optimal rate-information tradeoff. Finally, our results are corroborated by numerical examples.