Non-linear Information Inequalities

We construct non-linear information inequalities from Mat´uˇs’ infinite series of linear information inequalities. Each single non-linear inequality is sufficiently strong to prove that the closure of the set of all entropy functions is not polyhedral for four or more random variables, a fact that was already established using the series of linear inequalities. To the best of our knowledge, they are the first non-trivial examples of non-linear information inequalities.

[1]  Zhen Zhang,et al.  On Characterization of Entropy Function via Information Inequalities , 1998, IEEE Trans. Inf. Theory.

[2]  Lihua Song,et al.  Zero-error network coding for acyclic network , 2003, IEEE Trans. Inf. Theory.

[3]  Raymond W. Yeung,et al.  A framework for linear information inequalities , 1997, IEEE Trans. Inf. Theory.

[4]  Randall Dougherty,et al.  Six New Non-Shannon Information Inequalities , 2006, 2006 IEEE International Symposium on Information Theory.

[5]  Raymond W. Yeung,et al.  On a relation between information inequalities and group theory , 2002, IEEE Trans. Inf. Theory.

[6]  Frantisek Matús,et al.  Infinitely Many Information Inequalities , 2007, 2007 IEEE International Symposium on Information Theory.

[7]  Alex J. Grant,et al.  Dualities Between Entropy Functions and Network Codes , 2008, IEEE Transactions on Information Theory.