Browsing by Author "Kahle, Thomas"

Now showing 1 - 4 of 4
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    Algebraic Statistics
    (Zürich : EMS Publ. House, 2017) Kahle, Thomas; Sturmfels, Bernd; Uhler, Caroline
    Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research.
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    Algebraic Structures in Statistical Methodology
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2022) Drton, Mathias; Kahle, Thomas; Sullivant, Seth; Uhler, Caroline
    Algebraic structures arise naturally in a broad variety of statistical problems, and numerous fruitful connections have been made between algebra and discrete mathematics and research on statistical methodology. The workshop took up this theme with a particular focus on algebraic approaches to graphical models, causality, axiomatic systems for independence and non-parametric models.
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    Linear syzygies, hyperbolic Coxeter groups and regularity
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Constantinescu, Alexandru; Kahle, Thomas; Varbaro, Matteo
    We build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.
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    Positive margins and primary decomposition
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Kahle, Thomas; Rauh, Johannes; Sullivant, Seth
    We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the N-cycle and the complete bipartite graph K2,N2−2, with various restrictions on the size of the nodes.