An extremal inequality for long Markov chains

Let X, Y be jointly Gaussian vectors, and consider random variables U, V that satisfy the Markov constraint U - X - Y - V. We prove an extremal inequality relating the mutual informations between all (42) pairs of random variables from the set (U, X, Y, V). As a first application, we show that the rate region for the two-encoder quadratic Gaussian source coding problem follows as an immediate corollary of the the extremal inequality. In a second application, we establish the rate region for a vector-Gaussian source coding problem where Löwner-John ellipsoids are approximated based on rate-constrained descriptions of the data.

[1]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[2]  C. Porco.  From the blogosphere , 2007, Nature.

[3]  M. Henk.  Löwner – John Ellipsoids , 2012 .

[4]  B. L. S. Prakasa Rao,et al.  On a characterization of multivariate normal distribution , 1974 .

[5]  Pramod Viswanath,et al.  Rate Region of the Quadratic Gaussian Two-Encoder Source-Coding Problem , 2006, ISIT.

[6]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[7]  R. Durrett.  Probability: Theory and Examples , 1993 .

[8]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[9]  Pramod Viswanath,et al.  Rate Region of the Quadratic Gaussian Two-Encoder Source-Coding Problem , 2005, IEEE Transactions on Information Theory.

[10]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[11]  Tsachy Weissman,et al.  Multiterminal Source Coding Under Logarithmic Loss , 2011, IEEE Transactions on Information Theory.

[12]  Shashi M. Srivastava,et al.  A Course on Borel Sets , 1998, Graduate texts in mathematics.

[13]  Chandra Nair,et al.  The Capacity Region of the Two-Receiver Gaussian Vector Broadcast Channel With Private and Common Messages , 2014, IEEE Transactions on Information Theory.

[14]  Yasutada Oohama.  Gaussian multiterminal source coding , 1997, IEEE Trans. Inf. Theory.

[15]  Tsachy Weissman,et al.  On an Extremal Data Processing Inequality for long Markov Chains , 2014 .

[16]  M. Henk.  L¨ owner-John Ellipsoids , 2012 .