A recursive construction of the set of binary entropy vectors

Abstract

The primary contribution is a finite terminating algorithm that determines membership of a candidate entropy vector in the set of binary entropy vectors Φ N . We outline the relationship between Φ N and its unbounded cardinality discrete random variable counterpartΓ * N (or its normalizationΩ * N ). We discuss connections between Φ N andΩ * N . For example, for any outer bound, say the Shannon outer bound P N , toΩ * N , we provide a finite terminating algorithm to find a polytopic inner bound on Ω * N that agrees on tight faces of the outer bound.

Mapping the Region of Entropic Vectors with Support Enumeration & Information Geometry

“…Shifting from outer bounds to bounding from the inside, the most common way to generate inner bounds for the region of entropic vectors is to consider special families of distributions for which the entropy function is known to have certain properties. [15], [18]- [20] focus on calculating inner bounds based on special properties of binary random variables. However, the most common way to generate inner bounds is based on inequalities for representable matroids [21], boolean polymatroids [22], [23] and subspace arrangements.…”

“…The group action (17) on set partition and group action (18) on sets of set partitions enable us to enumerate the non-isomorphic k-atom supports by calculating the orbit data structure of symmetry group acting on a well defined set, the result is summarized in Theorem 2.…”