This project develops techniques for finding non-Shannon information inequalities and strives to geometrically characterize the space of vectors whose components are the entropies of subsets of jointly distributed discrete random variables. Specific sub-topics include the geometric description of inner and outer entropic region bounds, such as: the Shannon space, the Ingleton space, the common information space, the linearly representable space, and the copy region space. In addition, a unified approach to many important mathematical questions with engineering applications is studied using information theoretic inequality implications. Special cases include: random variable conditional independence implications, entropic region membership, information inequality identification, and network coding capacity bounding.
The problems in this field are of interest as fundamental theoretical questions and are also strongly motivated by communications applications in the area of network coding. Most of the study is analytic in nature, but the project also exploits computer assisted discovery and verification of new non-Shannon information inequalities for four and more random variables using techniques based upon auxiliary variables.
The discovery of new non-Shannon information inequalities provides improved bounds on the set of all entropic points. Such improved bounds lead to better bounds on the transmission capacity of networks. Calculating the capacity of a network is traditionally an extremely difficult, and usually intractible, problem. New inequalities also shed light on the basic understanding of both probability and information theory fundamentals. As information theory has proven to be an important and practical field over the last half century, a firm theoretical basis for it is necessary.
There are some broader impacts resulting from the proposed activity: The project involves active participation of graduate students, undergraduates, and also high school students, commensurate with their backgrounds and abilities. Graduate student participation provides training for careers in academia and industry. Undergraduate participation encourages research as a career and steers students towards academia and research in general.
