A Geometric Property of Relative Entropy and the Universal Threshold Phenomenon for Binary-Input Channels with Noisy State Information at the Encoder

TL;DR

This paper establishes bounds on the ratio of relative entropies for collinear distributions and applies these results to determine the universal noise threshold at which encoder side information no longer improves the capacity of binary-input channels.

Contribution

It introduces new bounds on relative entropy ratios for collinear distributions and uses them to identify the universal noise threshold in channels with noisy encoder side information.

Findings

Derived tight bounds on relative entropy ratios for collinear distributions.

Identified the exact universal noise threshold for encoder side information impact.

Analyzed capacity behavior of binary-input channels with noisy side information.

Abstract

Tight lower and upper bounds on the ratio of relative entropies of two probability distributions with respect to a common third one are established, where the three distributions are collinear in the standard $(n-1)$-simplex. These bounds are leveraged to analyze the capacity of an arbitrary binary-input channel with noisy causal state information (provided by a side channel) at the encoder and perfect state information at the decoder, and in particular to determine the exact universal threshold on the noise measure of the side channel, above which the capacity is the same as that with no encoder side information.