Strong Data Processing Inequalities for Input Constrained Additive Noise Channels

TLDR: In this article, it was shown that for the additive Gaussian noise channel with quadratic cost constraint, it is necessary and sufficient to saturate the channel to the point where mutual information is close to the maximum possible.

Abstract: This paper quantifies the intuitive observation that adding noise reduces available information by means of nonlinear strong data processing inequalities. Consider the random variables $W\to X\to Y$ forming a Markov chain, where $Y = X + Z$ with $X$ and $Z$ real valued, independent and $X$ bounded in $L_{p}$ -norm. It is shown that $I(W; Y) \le F_{I}(I(W;X))$ with $F_{I}(t) whenever $t > 0$ , if and only if $Z$ has a density whose support is not disjoint from any translate of itself. A related question is to characterize for what couplings $(W, X)$ the mutual information $I(W; Y)$ is close to maximum possible. To that end we show that in order to saturate the channel, i.e., for $I(W; Y)$ to approach capacity, it is mandatory that $I(W; X)\to \infty $ (under suitable conditions on the channel). A key ingredient for this result is a deconvolution lemma which shows that postconvolution total variation distance bounds the preconvolution Kolmogorov–Smirnov distance. Explicit bounds are provided for the special case of the additive Gaussian noise channel with quadratic cost constraint. These bounds are shown to be order optimal. For this case, simplified proofs are provided leveraging Gaussian-specific tools such as the connection between information and estimation (I-MMSE) and Talagrand’s information-transportation inequality.