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arXiv:1511.07829 · IEEE Transactions on Information Theory

$E_{\gamma}$-Resolvability

Jingbo Liu, Paul Cuff, Sergio Verdú

Published Jan 1, 2016• 22 citations
arXivHTMLPDF

Abstract

The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper, we study Eγ-resolvability, in which total variation is replaced by the more general Eγdistance. A general one-shot achievability bound for the precision of such an approximation is developed. Let QX|Ube a random transformation, n be an integer, and E ∈ (0, +∞). We show that in the asymptotic setting where γ = exp(nE), a (nonnegative) randomness rate above inf QU:D(QXIIπX)≤E{D(QXIIπX) + I(QU, QX|U) - E} is sufficient to approximate the output distribution πX⊗nusing the channel QX|U⊗n, where QU→ QX|U→ QX, and is also necessary in the case of finite U and X . In particular, a randomness rate of inf QUI(QU, QX|U) - E is always sufficient. We also study the convergence of the approximation error under the high-probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating Eγand other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth Rényi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive: 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d. setting; 2) a one-shot version of the mutual covering lemma; and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.

DOI
10.1109/tit.2016.2642111

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EγE_{\gamma}Eγ​-Resolvability
The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper, we study Eγ-resolvability, in which total variation is replaced by the more general Eγdistance. A general one-shot achievability bound for the precision of such an approximation is developed. Let QX|Ube a random transformation, n be an integer, and E ∈ (0, +∞). We show that in the asymptotic setting where γ = exp(nE), a (nonnegative) randomness rate above inf QU:D(QXIIπX)≤E{D(QXIIπX) + I(QU, QX|U) - E} is sufficient to approximate the output distribution πX⊗nusing the channel QX|U⊗n, where QU→ QX|U→ QX, and is also necessary in the case of finite U and X . In particular, a randomness rate of inf QUI(QU, QX|U) - E is always sufficient. We also study the convergence of the approximation error under the high-probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating Eγand other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth Rényi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive: 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d. setting; 2) a one-shot version of the mutual covering lemma; and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.
We study the amount of randomness needed for an input process to approximate a given output distribution of a channel in the Eγ distance. A general one-shot achievability bound for the precision of such an approximation is developed. In the i.i.d. setting where γ = exp(nE), a (nonnegative) randomness rate above infQU:D(QX||πX)≤E{D(QX||πX) + I(QU, QX|U) - E} is necessary and sufficient to asymptotically approximate the output distribution πX⊗nusing the channel QX|U⊗n, where QU→ QX|U→ QX. The new resolvability result is then used to derive a oneshot 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.
The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in[0, 2] U {∞}) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 1 as well as (lower and upper) bounds for the case when the same parameter is smaller than 1. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than 1, consistent with the traditional case where the Rényi parameter is equal to 1, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 1 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d. random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence. This tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information.
The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in [0,2]∪{∞}[0,2]\cup\{\infty\}[0,2]∪{∞}) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 111 as well as (lower and upper) bounds for the case when the same parameter is smaller than 111. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than~111, consistent with the traditional case where the Rényi parameter is equal to~111, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 111 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d.\ random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence.
We introduce the problem of variable-length (VL) source resolvability, in which a given target probability distribution is approximated by encoding a VL uniform random number, and the asymptotically minimum average length rate of the uniform random number, called the VL resolvability, is investigated. We first analyze the VL resolvability with the variational distance as an approximation measure. Next, we investigate the case under the divergence as an approximation measure. When the asymptotically exact approximation is required, it is shown that the resolvability under two kinds of approximation measures coincides. We then extend the analysis to the case of channel resolvability, where the target distribution is the output distribution via a general channel due to a fixed general source as an input. The obtained characterization of channel resolvability is fully general in the sense that, when the channel is just an identity mapping, it reduces to general formulas for source resolvability. We also analyze the second-order VL resolvability.
We consider the exact channel synthesis problem. This problem concerns the determination of the minimum amount of information required to create exact correlation remotely when there is a certain rate of randomness shared by two terminals. This problem generalizes an existing approximate version, in which the generated joint distribution is required to be close to a target distribution under the total variation (TV) distance measure (instead being exactly equal to the target distribution). We provide single-letter inner and outer bounds on the admissible region of the shared randomness rate and the communication rate for the exact channel synthesis problem. These two bounds coincide for doubly symmetric binary sources. We observe that for such sources, the admissible rate region for exact channel synthesis is strictly included in that for the TV-approximate version. We also extend the exact and TV-approximate channel synthesis problems to sources with countably infinite alphabets and continuous sources; the latter includes Gaussian sources. As by-products, lemmas concerning soft-covering under Rényi divergence measures are derived.
We consider the exact channel synthesis problem. This problem concerns the determination of the minimum amount of information required to create exact correlation remotely when there is a certain rate of randomness shared by two terminals. This problem generalizes an existing approximate version, in which the generated joint distribution is required to be close to a target distribution under the total variation (TV) distance measure (instead being exactly equal to the target distribution). We provide single-letter inner and outer bounds on the admissible region of the shared randomness rate and the communication rate for the exact channel synthesis problem. These two bounds coincide for doubly symmetric binary sources. We observe that for such sources, the admissible rate region for exact channel synthesis is strictly included in that for the TV-approximate version. We also extend the exact and TV-approximate channel synthesis problems to sources with countably infinite alphabets and continuous sources; the latter includes Gaussian sources. As by-products, lemmas concerning soft-covering under Rényi divergence measures are derived.