Advanced Geometric Algorithms Based on Discrete Analysis of Information-Geometric Structures with Applications

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Information objects such as Web graphs, quantum information, and knowledge are represented by manifolds in geometric spaces with coordinates naturally introduced by their associated numerical values. Information geometry investigates geometric structures of such spaces., while there have been little research on discrete algorithms working in the spaces. This project first investigates discrete geometric structures of information geometry, including quantum information geometry, and then develops efficient algorithms working in the space. Some topological properties such as shelling/orientation of polyhedral complexes and Web link graph structures are also investigated together with computational algebraic tools such as toxic ideals and their Grobner bases.Concerning Web structures, information compression techniques are developed by using discrete structure of the graph with regard to information entropy in the space. Topological extensions of shellings of a simplicial complex are made for oriented matroics, with connection to the realizability issue in real space.Discrete optimization algorithms have been devised in quantum information geometry, such as the Voronoi diagram of quantum states with respect to quantum divergence, minimum enclosing sphere of quantum states in connection with computing the Holevo capacity of a quantum communication channel. As one of remarkable results, we show the existence of a 4-signal 1-qubit quantum channel for the first time. Quantum entanglements and their hierarchical structures are also studied from the geometric viewpoint. With these results such new approaches to quantum information from discrete geometry have been shown to be a promising field.