@article{1833,
abstract = {Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in the table, and not necessarily containing the overall effect, that is, a common parameter in every cell. Similarly to log-linear models, relational models can be extended to non-negative distributions, but the extension requires more complex methods. An extended relational model is defined as an algebraic variety, and it turns out to be the closure of the original model with respect to the Bregman divergence. In the extended relational model, the MLE of the cell parameters always exists and is unique, but some of its properties may be different from those of the MLE under log-linear models. The MLE can be computed using a generalized iterative scaling procedure based on Bregman projections. },
author = {Klimova, Anna and Rudas, Tamás},
journal = {Journal of Multivariate Analysis},
pages = {440 -- 452},
publisher = {Elsevier},
title = {{On the closure of relational models}},
doi = {10.1016/j.jmva.2015.10.005},
volume = {143},
year = {2016},
}
@article{1480,
abstract = {Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials. },
author = {Michałek, Mateusz and Sturmfels, Bernd and Uhler, Caroline and Zwiernik, Piotr},
journal = {Proceedings of the London Mathematical Society},
number = {1},
pages = {27 -- 56},
publisher = {Oxford University Press},
title = {{Exponential varieties}},
doi = {10.1112/plms/pdv066},
volume = {112},
year = {2016},
}
@article{2006,
abstract = {The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed in our data. },
author = {Hein, Nicolas and Hillar, Christopher and Martin Del Campo Sanchez, Abraham and Sottile, Frank and Teitler, Zach},
journal = {Experimental Mathematics},
number = {3},
pages = {261 -- 269},
publisher = {Taylor & Francis},
title = {{The monotone secant conjecture in the real Schubert calculus}},
doi = {10.1080/10586458.2014.980044},
volume = {24},
year = {2015},
}
@article{1547,
abstract = {Let G be a graph on the vertex set V(G) = {x1,…,xn} with the edge set E(G), and let R = K[x1,…, xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials xixj with {xi,xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈Cxi for all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂ V(G) such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers LG and we explicitly describe the minimal free resolution of the ideal associated to LG which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.},
author = {Mohammadi, Fatemeh and Moradi, Somayeh},
issn = {2234-3016},
journal = {Bulletin of the Korean Mathematical Society},
number = {3},
pages = {977 -- 986},
publisher = {Korean Mathematical Society},
title = {{Resolution of unmixed bipartite graphs}},
doi = {10.4134/BKMS.2015.52.3.977},
volume = {52},
year = {2015},
}
@article{2014,
abstract = {The concepts of faithfulness and strong-faithfulness are important for statistical learning of graphical models. Graphs are not sufficient for describing the association structure of a discrete distribution. Hypergraphs representing hierarchical log-linear models are considered instead, and the concept of parametric (strong-) faithfulness with respect to a hypergraph is introduced. Strong-faithfulness ensures the existence of uniformly consistent parameter estimators and enables building uniformly consistent procedures for a hypergraph search. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. Lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness are computed for different parameterizations and measures of association.},
author = {Klimova, Anna and Uhler, Caroline and Rudas, Tamás},
journal = {Computational Statistics & Data Analysis},
number = {7},
pages = {57 -- 72},
publisher = {Elsevier},
title = {{Faithfulness and learning hypergraphs from discrete distributions}},
doi = {10.1016/j.csda.2015.01.017},
volume = {87},
year = {2015},
}