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.
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.
In literature, two different common informations were defined by G\'acs and K\orner and by Wyner, respectively. In this paper, we define a generalized version of common information, information-correlation function, by exploiting conditional maximal correlation as a commonness or privacy measure, which encompasses the notions of G\'acs-K\orner's and Wyner's common informations as special cases. Furthermore, we also study the problems of common information extraction and private sources synthesis, and show that information-correlation function is the optimal rate under a given conditional maximal correlation constraint for the centralized setting versions of these problems. As a side product, some properties on conditional maximal correlation have been derived as well.
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\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.
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.
We presented assisted common information as a generalization of Gacs-Korner (GK) common information at last year's ISIT. The motivation for our formulation was to improve upperbounds on the efficiency of protocols for secure two-party sampling (which is a form of secure multi-party computation). Our upperbound was based on a monotonicity property of a rate region (called the assisted residual information region) associated with the assisted common information formulation. In this note we present further results. We explore the connection of assisted common information with the Gray-Wyner system. We show that the assisted residual information region and the Gray-Wyner region are connected by a simple relationship: the assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map. Several known relationships between GK common information and Gray-Wyner system fall out as consequences of this. Quantities which arise in other source coding contexts acquire new interpretations. In previous work we showed that assisted common information can be used to derive upperbounds on the rate at which a pair of parties can securely sample correlated random variables, given correlated random variables from another distribution. Here we present an example where the bound derived using assisted common information is much better than previously known bounds, and in fact is tight. This example considers correlated random variables defined in terms of standard variants of oblivious transfer, and is interesting on its own as it answers a natural question about these cryptographic primitives.
Abstract—We presented assisted common information as a generalization of Gács-Körner (GK) common information at ISIT 2010. The motivation for our formulation was to improve upperbounds on the efficiency of protocols for secure two-party sampling (which is a form of secure multi-party computation). Our upperbound was based on a monotonicity property of a rate-region (called the assisted residual information region) associated with the assisted common information formulation. In this note we present further results. We explore the connection of assisted common information with the Gray-Wyner system. We show that the assisted residual information region and the Gray-Wyner region are connected by a simple relationship: the assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map. Several known relationships between GK common information and Gray-Wyner system fall out as consequences of this. Quantities which arise in other source coding contexts acquire new interpretations. In previous work we showed that assisted common information can be used to derive upperbounds on the rate at which a pair of parties can securely sample correlated random variables, given correlated random variables from another distribution. Here we present an example where the bound derived using assisted common information is much better than previously known bounds, and in fact is tight. This example considers correlated random variables defined in terms of standard variants of oblivious transfer, and is interesting on its own as it answers a natural question about these cryptographic primitives. I.