@article{1089,
abstract = {We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.},
author = {Fallat, Shaun and Lauritzen, Steffen and Sadeghi, Kayvan and Uhler, Caroline and Wermuth, Nanny and Zwiernik, Piotr},
issn = {00905364},
journal = {Annals of Statistics},
number = {3},
pages = {1152 -- 1184},
publisher = {Institute of Mathematical Statistics},
title = {{Total positivity in Markov structures}},
doi = {10.1214/16-AOS1478},
volume = {45},
year = {2017},
}
@article{1168,
abstract = {Optimum experimental design theory has recently been extended for parameter estimation in copula models. The use of these models allows one to gain in flexibility by considering the model parameter set split into marginal and dependence parameters. However, this separation also leads to the natural issue of estimating only a subset of all model parameters. In this work, we treat this problem with the application of the (Formula presented.)-optimality to copula models. First, we provide an extension of the corresponding equivalence theory. Then, we analyze a wide range of flexible copula models to highlight the usefulness of (Formula presented.)-optimality in many possible scenarios. Finally, we discuss how the usage of the introduced design criterion also relates to the more general issue of copula selection and optimal design for model discrimination.},
author = {Perrone, Elisa and Rappold, Andreas and Müller, Werner},
journal = {Statistical Methods and Applications},
number = {3},
pages = {403 -- 418},
publisher = {Springer},
title = {{D inf s optimality in copula models}},
doi = {10.1007/s10260-016-0375-6},
volume = {26},
year = {2017},
}
@article{1208,
abstract = {We study parameter estimation in linear Gaussian covariance models, which are p-dimensional Gaussian models with linear constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local maxima. Using recent results on the asymptotic distribution of extreme eigenvalues of the Wishart distribution, we provide sufficient conditions for any hill climbing method to converge to the global maximum. Although we are primarily interested in the case in which n≫p, the proofs of our results utilize large sample asymptotic theory under the scheme n/p→γ>1. Remarkably, our numerical simulations indicate that our results remain valid for p as small as 2. An important consequence of this analysis is that, for sample sizes n≃14p, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem. © 2016 The Royal Statistical Society and Blackwell Publishing Ltd.},
author = {Zwiernik, Piotr and Uhler, Caroline and Richards, Donald},
issn = {13697412},
journal = {Journal of the Royal Statistical Society. Series B: Statistical Methodology},
number = {4},
pages = {1269 -- 1292},
publisher = {Wiley-Blackwell},
title = {{Maximum likelihood estimation for linear Gaussian covariance models}},
doi = {10.1111/rssb.12217},
volume = {79},
year = {2017},
}
@article{698,
abstract = {Extracellular matrix signals from the microenvironment regulate gene expression patterns and cell behavior. Using a combination of experiments and geometric models, we demonstrate correlations between cell geometry, three-dimensional (3D) organization of chromosome territories, and gene expression. Fluorescence in situ hybridization experiments showed that micropatterned fibroblasts cultured on anisotropic versus isotropic substrates resulted in repositioning of specific chromosomes, which contained genes that were differentially regulated by cell geometries. Experiments combined with ellipsoid packing models revealed that the mechanosensitivity of chromosomes was correlated with their orientation in the nucleus. Transcription inhibition experiments suggested that the intermingling degree was more sensitive to global changes in transcription than to chromosome radial positioning and its orientations. These results suggested that cell geometry modulated 3D chromosome arrangement, and their neighborhoods correlated with gene expression patterns in a predictable manner. This is central to understanding geometric control of genetic programs involved in cellular homeostasis and the associated diseases. },
author = {Wang, Yejun and Nagarajan, Mallika and Uhler, Caroline and Shivashankar, Gv},
issn = {10591524},
journal = {Molecular Biology of the Cell},
number = {14},
pages = {1997 -- 2009},
publisher = {American Society for Cell Biology},
title = {{Orientation and repositioning of chromosomes correlate with cell geometry dependent gene expression}},
doi = {10.1091/mbc.E16-12-0825},
volume = {28},
year = {2017},
}
@article{1293,
abstract = {For a graph G with p vertices the closed convex cone S⪰0(G) consists of all real positive semidefinite p×p matrices whose sparsity pattern is given by G, that is, those matrices with zeros in the off-diagonal entries corresponding to nonedges of G. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. While the maximum rank of an extremal ray in S⪰0(G), known as the sparsity order of G, has been characterized for different classes of graphs, we here study all possible extremal ranks of S⪰0(G). We investigate when the geometry of the (±1)-cut polytope of G yields a polyhedral characterization of the set of extremal ranks of S⪰0(G). For a graph G without K5 minors, we show that appropriately chosen normal vectors to the facets of the (±1)-cut polytope of G specify the off-diagonal entries of extremal matrices in S⪰0(G). We also prove that for appropriately chosen scalars the constant term of the linear equation of each facet-supporting hyperplane is the rank of its corresponding extremal matrix in S⪰0(G). Furthermore, we show that if G is series-parallel then this gives a complete characterization of all possible extremal ranks of S⪰0(G). Consequently, the sparsity order problem for series-parallel graphs can be solved in terms of polyhedral geometry.},
author = {Solus, Liam T and Uhler, Caroline and Yoshida, Ruriko},
journal = {Linear Algebra and Its Applications},
pages = {247 -- 275},
publisher = {Elsevier},
title = {{Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors}},
doi = {10.1016/j.laa.2016.07.026},
volume = {509},
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{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{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{1579,
abstract = {We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.},
author = {Brooks, Christopher and Martin Del Campo Sanchez, Abraham and Sottile, Frank},
journal = {Transactions of the American Mathematical Society},
number = {6},
pages = {4183 -- 4206},
publisher = {American Mathematical Society},
title = {{Galois groups of Schubert problems of lines are at least alternating}},
doi = {10.1090/S0002-9947-2014-06192-8},
volume = {367},
year = {2015},
}
@article{1997,
abstract = {We prove that the three-state toric homogeneous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals is generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by degree six binomials.},
author = {Noren, Patrik},
journal = {Journal of Symbolic Computation},
number = {May-June},
pages = {285 -- 296},
publisher = {Elsevier},
title = {{The three-state toric homogeneous Markov chain model has Markov degree two}},
doi = {10.1016/j.jsc.2014.09.014},
volume = {68/Part 2},
year = {2015},
}