The Entropic Doubling Constant and Robustness of Gaussian Codebooks for Additive-Noise Channels

Entropy comparison inequalities are obtained for the differential entropy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h(X+Y)$ </tex-math></inline-formula> of the sum of two independent random vectors <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X,Y$ </tex-math></inline-formula>, when one is replaced by a Gaussian. For identically distributed random vectors <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X,Y$ </tex-math></inline-formula>, these are closely related to bounds on the entropic doubling constant, which quantifies the entropy increase when adding an independent copy of a random vector to itself. Consequences of both large and small doubling are explored. For the former, lower bounds are deduced on the entropy increase when adding an independent Gaussian, while for the latter, a qualitative stability result for the entropy power inequality is obtained. In the more general case of non-identically distributed random vectors <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X,Y$ </tex-math></inline-formula>, a Gaussian comparison inequality with interesting implications for channel coding is established: For additive-noise channels with a power constraint, Gaussian codebooks come within a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {\mathsf { snr}}{3{\mathsf { snr}}+2}$ </tex-math></inline-formula> factor of capacity. In the low-SNR regime this improves the half-a-bit additive bound of Zamir and Erez. Analogous results are obtained for additive-noise multiple access channels, and for linear, additive-noise