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On the Theory of Polynomial Information Inequalities On the theory of polynomial information inequalities Arley Rams´esG´omezR´ıos Universidad Nacional de Colombia Facultad de Ciencias, Departamento de Matem´aticas Bogot´a,Colombia 2018 On the theory of polynomial information inequalities Arley Rams´esG´omezR´ıos Dissertation submitted to the Department of Mathematics in partial fulfilment of the requirements for the degrees of Doctor of Philosophy in Mathematics Advisor: Juan Andr´esMontoya. Ph.D. Research Topic: Information Theory Universidad Nacional de Colombia Facultad de Ciencias, Departamento de Matem´aticas Bogot´a,Colombia 2018 Approved by |||||||||||||||||| Professor Laszlo Csirmaz |||||||||||||||||| Professor Humberto Sarria Zapata |||||||||||||||||| Professor Jorge Eduardo Ortiz Acknowledgement I would like to express my deep gratitude to Professor Juan Andres Montoya, for proposing this research topic, for all the advice and suggestions, for contributing his knowledge, and for the opportunity to work with him. It has been a great experience. I would also like to thank my parents Ismael G´omezand Livia R´ıos,for their unconditional support throughout my study, my girlfriend for her encouragement and my brothers who support me in every step I take. My deepest thanks are also extended to my research partner Carolina Mej´ıa,and all the people involved: researchers, colleagues and friends who in one way or an- other contributed to the completion of this work. Finally I wish to thank the National University of Colombia, for my professional training and for the financial support during the development of this research work. v Resumen En este trabajo estudiamos la definibilidad de las regiones cuasi entr´opicaspor medio de conjuntos finitos de desigualdades polinomiales. Los conjuntos que son definidos de esta manera son llamados semialgebraicos. Existe una fuerte conexi´onentre los conjuntos semialgebraicos y la Teor´ıade Modelos, esta conexi´onse presenta a trav´es del llamado teorema de Tarski Seidenberg. Nosotros exploramos esta conexi´on,por ejemplo, probamos que el conjunto de vectores entr´opicosde orden mayor a dos no es semialgebraico, y presentamos resultados que sugieren que las regiones cuasi entr´opicasde orden mayor a tres no son semialgebraicas. Primero presentamos una prueba alternativa del teorema de Mat´uˇs,el cual afirma que las regiones cuasi entr´opicasno son poli´edricas,despu´esabordamos el problema de encontrar nuevas sucesiones de desigualdades de la informaci´ony finalmente mostramos que la semial- gebricidad de las regiones cuasi entr´opicasdepende de la condicionalidad esencial de cierta clase de desigualdades condicionales de la informaci´on.Exploramos adem´as algunas consecuencias algor´ıtmicasque podr´ıatener el hecho de que las regiones cuasi entr´opicasfuesen semialgebraicas, espec´ıficamente estudiamos algunas con- secuencias en la Teor´ıade Repartici´onde Secretos y su relaci´oncon la Teor´ıade Matroides. Palabras clave: Entrop´ıa, Vectores entr´opicos,Desigualdades de la infor- maci´on,Regiones entr´opicas,Repartici´onde secretos. vi Abstract We study the definability of the almost entropic regions by finite sets of polynomial inequalities. Sets defined in this way are called semialgebraic. There is a strong con- nection between semialgebraic sets and Model Theory, this connection is presented through the so-called Tarski-Seidenberg Theorem. We explore this connection and, for instance, we prove that the set of entropic vectors of order greater than two is not semialgebraic. Moreover, we present strong evidence suggesting that the almost entropic regions of order greater than three are not semialgebraic. First we present an alternative proof of Mat´uˇstheorem, which states that the almost entropic re- gions are not polyhedral, then we deal with the problem of finding new sequences of information inequalities and finally we show that the semialgebraicity of the almost entropic regions depends on the essential conditionality of certain class of condi- tional information inequalities. We also explore some algorithmic consequences of the almost entropic regions being semialgebraic, specifically we study some of the consequences of this fact in Secret Sharing and its relation with Matroid Theory. Keywords: Entropy, Entropic Vectors, Information inequalities, Entropic regions, Secret Sharing. vii Publications related to this thesis 1. A. Gomez, C. Mejia, and J.A. Montoya. Linear network coding and the model theory of linear rank inequalities. IEEE International Symposium on Network Coding (NetCod) (Aalborg, Denmark), (2014), 1 - 6. 2. A. Gomez, C. Mejia, and J.A. Montoya. Network coding and the model theory of linear information inequalities. IEEE International Symposium on Network Coding (NetCod) (Aalborg, Denmark), (2014), 1 - 6. 3. A. Gomez, C. Mejia and J.A. Montoya. Defining the almost entropic regions by algebraic inequalities. International Journal of Information and Coding Theory (IJICOT), (2017) Vol 4, No. 1, 1 - 18. 4. A. Gomez, C. Mejia and J.A. Montoya. On the Linear Shareability of Ma- troids. Applied Mathematical Sciences, (2017), Vol 11, No. 48, 2351-2365. Citations: 1. Joseph Connelly and Kenneth Zeger. Linear Capacity of Networks over Ring Alphabets. arXiv Preprints, arXiv: 1706.01152 (2017). viii 1 Contents Acknowledgement iv Abstract v Introduction xi 0.1 Organization of the work . xiv 1 Basics 1 1.1 Entropy and Shannon Information Measures . .1 1.2 Information inequalities and entropic vectors . .4 1.3 Shannon Information Inequalities . .9 1.4 Linear Rank Inequalities . 11 1.4.1 Ingleton inequality . 14 2 Semialgebraicity of the almost entropic regions 17 2.1 Non-linear information inequalities and semialgebraic sets . 17 2.2 Tarski-Seidenberg Theorem . 21 2.3 The almost entropic region is defined by a single nonlinear inequality 24 3 The importance of being semialgebraic, a Secret Sharing approach 29 3.1 Secret Sharing . 30 x CONTENTS 3.1.1 Rates . 33 3.2 Ideal access structures . 34 3.3 Access structures from matroids . 36 3.3.1 Matroids . 36 3.3.2 Ideal matroids . 39 3.4 Characterizing the class of Ideal Matroids . 40 4 Disproving semialgebraicity 48 4.1 An appropriate basis for R15 ....................... 48 4.2 Mat´uˇstheorem revisited . 49 4.3 A two-dimensional view . 55 ∗ ◦ 4.4 Looking for a nonsemialgebraic two-dimensional section of Γ4 ... 59 4.5 Looking for new sequences: A Linear Algebra approach . 60 4.5.1 Copy Lemma . 60 4.5.2 Linear invariants and extension rules . 64 4.6 A good old sequence . 71 5 Concluding remarks and questions for future research 77 Bibliography 79 Introduction Linear Information Inequalities are the linear Inequalities satisfied by Shannon's Entropy. N. Pippenger [35] argued that linear information inequalities encode the fundamental laws of Information Theory, which determine the limits of information transmission and data compression. Then, according to Pippenger, those inequali- ties constitute a very important topic of research. Linear information inequalities play an important role in the analysis of communi- cations problems. Unfortunately, it is not easy to decide if a given linear expression involving Shannon Entropies is always positive. Actually, it is not known if the set of linear information inequalities is a decidible set. R. Yeung introduced a geometrical framework to study the theory of linear infor- mation inequalities [47]. To this end he introduced the notions of entropic vector, entropic region, and almost entropic region. The almost entropic region of order n is precisely the set that is defined by all linear information inequalities in n random variables. He proved, for instance, that the almost entropic regions are closed convex cones. A first example of information inequalities are the so called Shannon inequali- ties (claiming that all the Shannon information measures are positive). Are there non-Shannon information inequalities? That is: there do exist linear information inequalities that are not entailed by Shannon inequalities? It was conjectured for a xii 0 Introduction long time that there exist non-Shannon information inequalities. L. Csirmaz pub- lished an influential paper in 1997 [14], where he proved that Shannon inequalities do not yield super-linear lower bounds for secret sharing. This work of Csirmaz suggested that Shannon's inequalities are not a complete set of linear inequalities (axioms) defining the convex cones constituted by the entropic vectors of different orders. After that, an intensive search for non-Shannon inequalities began. Zhang and Yeung [49] found the first non-Shannon unconditional information inequality, and after that, Dougherty et al [16] found six new non-Shannon information in- equalities. Are those seven information inequalities (and their permutations) the lost axioms of Shannon's entropy? If it were the case, all the almost entropic re- gions would become polyhedral cones, and it would be good news, given that those regions appear as essential parameters of possible algorithmic solutions to many dif- ferent problems related to Network Coding [47], Secret Sharing[29], [14], database theory, [18], graph guessing [36] and Index coding [38]. Notice that Polyhedral cones are tractable objects. It is the case because they are defined by a finite list of linear inequalities, which can be algorithmically checked. Moreover, polyhedral cones can be processed using the many tools provided by convex geometry. We claim that solving all the aforementioned problems is something that strongly depends on our ability for computing finite checkable definitions
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