17,979 research outputs found

    Optimal Kullback-Leibler Aggregation via Information Bottleneck

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    In this paper, we present a method for reducing a regular, discrete-time Markov chain (DTMC) to another DTMC with a given, typically much smaller number of states. The cost of reduction is defined as the Kullback-Leibler divergence rate between a projection of the original process through a partition function and a DTMC on the correspondingly partitioned state space. Finding the reduced model with minimal cost is computationally expensive, as it requires an exhaustive search among all state space partitions, and an exact evaluation of the reduction cost for each candidate partition. Our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost; The proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition. Then, we express the problem in the form of information bottleneck optimization, and propose using the agglomerative information bottleneck algorithm for searching a sub-optimal partition greedily, rather than exhaustively. The theory is illustrated with examples and one application scenario in the context of modeling bio-molecular interactions.Comment: 13 pages, 4 figure

    Bottleneck Surfaces and Worldsheet Geometry of Higher-Curvature Quantum Gravity

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    We describe a simple lattice model of higher-curvature quantum gravity in two dimensions and study the phase structure of the theory as a function of the curvature coupling. It is shown that the ensemble of flat graphs is entropically unstable to the formation of baby universes. In these simplified models the growth in graphs exhibits a branched polymer behaviour in the phase directly before the flattening transition.Comment: 18 pages LaTeX, 3 .eps figures, uses epsf.tex; clarifying comments added and typos correcte

    Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

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    We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set PP of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle â–½\triangledown, and there is an edge between two points in PP if and only if there is an empty homothet of â–½\triangledown having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely kk-TD, which contains an edge between two points if the interior of the homothet of â–½\triangledown having the two points on its boundary contains at most kk points of PP. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of kk-TD. Finally we consider the problem of blocking the edges of kk-TD.Comment: 20 page
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