  The two most prevalent notions of common information (CI) are due to Wyner
and Gacs-Korner and both the notions can be stated as two different
characteristic points in the lossless Gray-Wyner region. Although the
information theoretic characterizations for these two CI quantities can be
easily evaluated for random variables with infinite entropy (eg., continuous
random variables), their operational significance is applicable only to the
lossless framework. The primary objective of this paper is to generalize these
two CI notions to the lossy Gray-Wyner network, which hence extends the
theoretical foundation to general sources and distortion measures. We begin by
deriving a single letter characterization for the lossy generalization of
Wyner's CI, defined as the minimum rate on the shared branch of the Gray-Wyner
network, maintaining minimum sum transmit rate when the two decoders
reconstruct the sources subject to individual distortion constraints. To
demonstrate its use, we compute the CI of bivariate Gaussian random variables
for the entire regime of distortions. We then similarly generalize Gacs and
Korner's definition to the lossy framework. The latter half of the paper
focuses on studying the tradeoff between the total transmit rate and receive
rate in the Gray-Wyner network. We show that this tradeoff yields a contour of
points on the surface of the Gray-Wyner region, which passes through both the
Wyner and Gacs-Korner operating points, and thereby provides a unified
framework to understand the different notions of CI. We further show that this
tradeoff generalizes the two notions of CI to the excess sum transmit rate and
receive rate regimes, respectively.
