Resolvability in Eγ with Applications to 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 ${\inf }_{Q_{\mathsf{U}}:D(Q_{\mathsf{X}}||{\pi }_{\mathsf{X}})\le E}\{D(Q_{\mathsf{X}}||{\pi }_{\mathsf{X}})+I(Q_{\mathsf{U}},Q_{\mathsf{X}|\mathsf{U}})-E\}$ is necessary and sufficient to asymptotically approximate the output distribution ${\pi }_{\mathsf{X}}^{\otimes n}$ using the channel $Q_{\mathsf{X}|\mathsf{U}}^{\otimes n}$, where $Q_{\mathsf{U}}\to Q_{\mathsf{X}|\mathsf{U}}\to Q_{\mathsf{X}}$. 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.

ISIT 2015

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