Book Inequalities

Abstract

Information theoretical inequalities have strong ties with polymatroids and their representability. A polymatroid is entropic if its rank function is given by the Shannon entropy of the subsets of some discrete random variables. The book is a special iterated adhesive extension of a polymatroid with the property that entropic polymatroids have n-page book extensions over an arbitrary spine. We prove that every polymatroid has an n-page book extension over a single element and over an allbut-one-element spine. Consequently, for polymatroids on four elements, only book extensions over a two-element spine should be considered. F. Matúš proved that the Zhang-Yeung inequalities characterize polymatroids on four elements which have such a 2-page book extension. The n-page book inequalities, defined in this paper, are conjectured to characterize polymatroids on four elements which have n-page book extensions over a two-element spine. We prove that the condition is necessary; consequently every book inequality is an information inequality on four random variables. Using computer-aided multiobjective optimization, the sufficiency of the condition is verified up to 9-page book extensions.

Entropy Region and Convolution

“…by (11). Hence, the score I ij (g) is approximately −0.09243, currently the best upper bound on the infimal Ingleton score.…”

“…Main breakthroughs include finding of the first nonShannon linear inequality by Zhang-Yeung [46] and the relation to group theory by [6]. The cone cl(H ent N ) is not polyhedral [35] and the structure of non-Shannon inequalities seems to be complex [45,28,13,14,11,43]. Reviews are in [9,34] and elsewhere.…”

Entropy region and convolution

“…The convex cone cl(H ent N ) is not polyhedral [35], thus its polar cone is not finitely generated. There are infinite sets of linear information-theoretic inequalities [45,28,13,14,11], some of them rigorously proved and hundreds of them generated in computer experiments. The experiments are based on the fact that the entropic polymatroids are selfadhesive, H ent N ⊆ H sa N , and iterations of the idea.…”

“…Main breakthroughs include finding of the first non-Shannon linear inequality by Zhang-Yeung [46] and the relation to group theory by [6]. The cone cl(H ent N ) is not polyhedral [35] and the structure of non-Shannon inequalities seems to be complex [45,28,13,14,11,43]. Reviews are in [9,34] and elsewhere.…”

How to Use Undiscovered Information Inequalities: Direct Applications of the Copy Lemma

“…). F. Matúš and L. Csirmaz proposed several extensions and generalizations of the Copy Lemma (polymatroid convolution, book extension, maximum entropy extension, see, e.g., [12], [22], [23]). However, to the best of our knowledge, the "classical" version of the Copy Lemma is strong enough for all known proofs of non-Shannon type inequalities All known proofs of (unconditional) non-Shannon type information inequalities can be presented in the following style: we start with a distribution (X 1 , .…”