Recently, two extensions of Wyner's common information-exact and Rényi common informations-were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems refers to determining the minimum rate of the common input to two independent processors needed to generate an exact or approximate joint distribution. For the exact common information problem, an exact generation of the target distribution is required, while for Wyner's and α-Rényi common informations, the relative entropy and Rényi divergence with order α were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner's common information. However, it was hitherto unknown whether the former is strictly larger than the latter. In this paper, we first establish the equivalence between the exact and ∞-Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and ∞-Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner's. This answers an open problem posed by KLE.
Recently, two extensions of Wyner's common information-exact and Rényi common informations-were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems refers to determining the minimum rate of the common input to two independent processors needed to generate an exact or approximate joint distribution. For the exact common information problem, an exact generation of the target distribution is required, while for Wyner's and α-Rényi common informations, the relative entropy and Rényi divergence with order α were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner's common information. However, it was hitherto unknown whether the former is strictly larger than the latter. In this paper, we first establish the equivalence between the exact and ∞-Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and ∞-Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner's. This answers an open problem posed by KLE.
Recently, two extensions of Wyner's common information-exact and Rényi common informations-were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution. For the exact common information problem, exact generation of the target distribution is required, while for Wyner's and α-Rényi common informations, the relative entropy and Rényi divergence with order α were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner's common information. However, it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions. In this paper, we first establish the equivalence between the exact and ∞-Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and ∞-Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner's. This answers an open problem posed by KLE. Furthermore, we extend Wyner's, ∞-Rényi, and exact common informations to sources with countably infinite or continuous alphabets, including Gaussian sources.
Recently, two extensions of Wyner's common information\textemdash exact and Rényi common informations\textemdash were introduced respectively by Kumar, Li, and El Gamal (KLE), and the present authors. The class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution. For the exact common information problem, exact generation of the target distribution is required, while for Wyner's and α-Rényi common informations, the relative entropy and Rényi divergence with order α were respectively used to quantify the discrepancy between the synthesized and target distributions. The exact common information is larger than or equal to Wyner's common information. However, it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions. In this paper, we first establish the equivalence between the exact and ∞-Rényi common informations, and then provide single-letter upper and lower bounds for these two quantities. For doubly symmetric binary sources, we show that the upper and lower bounds coincide, which implies that for such sources, the exact and ∞-Rényi common informations are completely characterized. Interestingly, we observe that for such sources, these two common informations are strictly larger than Wyner's. This answers an open problem posed by KLE. Furthermore, we extend Wyner's, ∞-Rényi, and exact common informations to sources with countably infinite or continuous alphabets, including Gaussian sources.
We study a generalized version of Wyner's common information problem (also coined the distributed source simulation problem). The original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and target distributions is the normalized relative entropy. Our generalization involves changing the distance measure to the unnormalized and normalized Rényi divergences of order α=1+s∈[0,2]. We show that the minimum rate needed to ensure the Rényi divergences between the distribution induced by a code and the target distribution vanishes remains the same as the one in Wyner's setting, except when the order α=1+s=0. This implies that Wyner's common information is rather robust to the choice of distance measure employed. As a byproduct of the proofs used to the establish the above results, the exponential strong converse for the common information problem under the total variation distance measure is established.
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.
This paper introduces the notion of exact common information, which is the minimum description length of the common randomness needed for the exact distributed generation of two correlated random variables (X, Y). We introduce the quantity G(X; Y) = minX→W→YH(W) as a natural bound on the exact common information and study its properties and computation. We then introduce the exact common information rate, which is the minimum description rate of the common randomness for the exact generation of a 2-DMS (X, Y). We give a multiletter characterization for it as the limit Ḡ(X; Y) = limn→∞(1/n)G(Xn; Yn). While in general Ḡ(X; Y) is greater than or equal to the Wyner common information, we show that they are equal for the Symmetric Binary Erasure Source. We do not know, however, if the exact common information rate has a single letter characterization in general.
In literature, different common informations were defined by Gács and Körner, by Wyner, and by Kumar, Li, and Gamal, respectively. In this paper, we define two generalized versions of common informations, named approximate and exact information-correlation functions, by exploiting the conditional maximal correlation as a commonness or privacy measure. These two generalized common informations encompass the notions of Gács-Körner's, Wyner's, and Kumar-Li-Gamal's common informations as special cases. Furthermore, to give operational characterizations of these two generalized common informations, we also study the problems of private sources synthesis and common information extraction, and show that the information-correlation functions are equal to the minimum rates of commonness needed to ensure that some conditional maximal correlation constraints are satisfied for the centralized setting versions of these problems. As a byproduct, the conditional maximal correlation has been studied as well.