Resolvability in E{\gamma} with Applications to Lossy Compression and Wiretap Channels

TL;DR

This paper investigates the amount of randomness required to approximate output distributions of channels using $E_{oldsymbol{ ext{gamma}}}$ distance, providing bounds applicable to lossy compression and wiretap channels, with both one-shot and asymptotic analyses.

Contribution

It develops a general one-shot achievability bound for distribution approximation and derives tight asymptotic bounds for resolvability, lossy compression, and wiretap channels.

Findings

Established a one-shot bound for distribution approximation accuracy.

Derived necessary and sufficient randomness rates in the i.i.d. setting.

Provided asymptotically tight bounds for lossy compression and wiretap channels.

Abstract

We study the amount of randomness needed for an input process to approximate a given output distribution of a channel in the $E_{\gamma}$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. In the i.i.d.~setting where $\gamma=\exp(nE)$, a (nonnegative) randomness rate above $\inf_{Q_{\sf U}: D(Q_{\sf X}||\pi_{\sf X})\le E} \{D(Q_{\sf X}||\pi_{\sf X})+I(Q_{\sf U},Q_{\sf X|U})-E\}$ is necessary and sufficient to asymptotically approximate the output distribution $\pi_{\sf X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$, where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$. The new resolvability result is then used to derive a one-shot upper bound on the error probability in the rate distortion problem, and a lower bound on the size of the eavesdropper list to include the actual message in the wiretap channel problem. Both bounds are asymptotically tight in i.i.d.~settings.