@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{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{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{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{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{1621,
author = {Hillar, Christopher and Martin Del Campo Sanchez, Abraham},
journal = {Journal of Symbolic Computation},
pages = {650 -- 652},
publisher = {Elsevier},
title = {{Corrigendum to “Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals” [J. Symb. Comput. 50 (March 2013) 314–334]}},
doi = {10.1016/j.jsc.2015.09.002},
volume = {74},
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},
}
@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},
}
@article{2008,
abstract = {The paper describes a generalized iterative proportional fitting procedure that can be used for maximum likelihood estimation in a special class of the general log-linear model. The models in this class, called relational, apply to multivariate discrete sample spaces that do not necessarily have a Cartesian product structure and may not contain an overall effect. When applied to the cell probabilities, the models without the overall effect are curved exponential families and the values of the sufficient statistics are reproduced by the MLE only up to a constant of proportionality. The paper shows that Iterative Proportional Fitting, Generalized Iterative Scaling, and Improved Iterative Scaling fail to work for such models. The algorithm proposed here is based on iterated Bregman projections. As a by-product, estimates of the multiplicative parameters are also obtained. An implementation of the algorithm is available as an R-package.},
author = {Klimova, Anna and Rudas, Tamás},
journal = {Scandinavian Journal of Statistics},
number = {3},
pages = {832 -- 847},
publisher = {Wiley},
title = {{Iterative scaling in curved exponential families}},
doi = {10.1111/sjos.12139},
volume = {42},
year = {2015},
}
@article{2013,
abstract = {An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC algorithm and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow ties, tripartite graphs, and complete graphs.
},
author = {Lin, Shaowei and Uhler, Caroline and Sturmfels, Bernd and Bühlmann, Peter},
journal = {Foundations of Computational Mathematics},
number = {5},
pages = {1079 -- 1116},
publisher = {Springer},
title = {{Hypersurfaces and their singularities in partial correlation testing}},
doi = {10.1007/s10208-014-9205-0},
volume = {14},
year = {2014},
}
@misc{2007,
author = {Anna Klimova and Rudas, Tamás},
publisher = {The Comprehensive R Archive Network},
title = {{gIPFrm: Generalized iterative proportional fitting for relational models}},
year = {2014},
}
@article{1911,
abstract = {The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Δ, where Δ is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>Δ 2 then the topological Tverberg theorem still works. It is conjectured that q>KΔ is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.},
author = {Engström, Alexander and Noren, Patrik},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {207 -- 220},
publisher = {Springer},
title = {{Tverberg's Theorem and Graph Coloring}},
doi = {10.1007/s00454-013-9556-3},
volume = {51},
year = {2014},
}
@article{2178,
abstract = {We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time T, the size of the design matrix is 6 × 3 · 2T-1 and the convex hull of its columns is the model polytope. We study the behavior of this polytope for T ≥ 3 and we show that it is defined by 24 facets for all T ≥ 5. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.},
author = {Haws, David and Martin Del Campo Sanchez, Abraham and Takemura, Akimichi and Yoshida, Ruriko},
journal = {Beitrage zur Algebra und Geometrie},
number = {1},
pages = {161 -- 188},
publisher = {Springer},
title = {{Markov degree of the three-state toric homogeneous Markov chain model}},
doi = {10.1007/s13366-013-0178-y},
volume = {55},
year = {2014},
}
@inproceedings{2047,
abstract = {Following the publication of an attack on genome-wide association studies (GWAS) data proposed by Homer et al., considerable attention has been given to developing methods for releasing GWAS data in a privacy-preserving way. Here, we develop an end-to-end differentially private method for solving regression problems with convex penalty functions and selecting the penalty parameters by cross-validation. In particular, we focus on penalized logistic regression with elastic-net regularization, a method widely used to in GWAS analyses to identify disease-causing genes. We show how a differentially private procedure for penalized logistic regression with elastic-net regularization can be applied to the analysis of GWAS data and evaluate our method’s performance.},
author = {Yu, Fei and Rybar, Michal and Uhler, Caroline and Fienberg, Stephen},
booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
editor = {Domingo Ferrer, Josep},
location = {Ibiza, Spain},
pages = {170 -- 184},
publisher = {Springer},
title = {{Differentially-private logistic regression for detecting multiple-SNP association in GWAS databases}},
doi = {10.1007/978-3-319-11257-2_14},
volume = {8744},
year = {2014},
}
@article{2011,
abstract = {The protection of privacy of individual-level information in genome-wide association study (GWAS) databases has been a major concern of researchers following the publication of “an attack” on GWAS data by Homer et al. (2008). Traditional statistical methods for confidentiality and privacy protection of statistical databases do not scale well to deal with GWAS data, especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach that provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees may come at a serious price in terms of data utility. Building on such notions, Uhler et al. (2013) proposed new methods to release aggregate GWAS data without compromising an individual’s privacy. We extend the methods developed in Uhler et al. (2013) for releasing differentially-private χ2χ2-statistics by allowing for arbitrary number of cases and controls, and for releasing differentially-private allelic test statistics. We also provide a new interpretation by assuming the controls’ data are known, which is a realistic assumption because some GWAS use publicly available data as controls. We assess the performance of the proposed methods through a risk-utility analysis on a real data set consisting of DNA samples collected by the Wellcome Trust Case Control Consortium and compare the methods with the differentially-private release mechanism proposed by Johnson and Shmatikov (2013).},
author = {Yu, Fei and Fienberg, Stephen and Slaković, Alexandra and Uhler, Caroline},
journal = {Journal of Biomedical Informatics},
pages = {133 -- 141},
publisher = {Elsevier},
title = {{Scalable privacy-preserving data sharing methodology for genome-wide association studies}},
doi = {10.1016/j.jbi.2014.01.008},
volume = {50},
year = {2014},
}
@inproceedings{2012,
abstract = {The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice.},
author = {Iglesias Ham, Mabel and Kerber, Michael and Uhler, Caroline},
location = {Halifax, Canada},
pages = {155 -- 161},
publisher = {Unknown},
title = {{Sphere packing with limited overlap}},
year = {2014},
}
@article{2280,
abstract = {The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application-chromosome organization in the human cell nucleus-is discussed briefly, and some illustrative results are presented.},
author = {Uhler, Caroline and Wright, Stephen},
journal = {SIAM Review},
number = {4},
pages = {671 -- 706},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Packing ellipsoids with overlap}},
doi = {10.1137/120872309},
volume = {55},
year = {2013},
}
@article{2010,
abstract = {Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.},
author = {Uhler, Caroline and Raskutti, Garvesh and Bühlmann, Peter and Yu, Bin},
journal = {The Annals of Statistics},
number = {2},
pages = {436 -- 463},
publisher = {Institute of Mathematical Statistics},
title = {{Geometry of the faithfulness assumption in causal inference}},
doi = {10.1214/12-AOS1080},
volume = {41},
year = {2013},
}
@article{2009,
abstract = {Traditional statistical methods for confidentiality protection of statistical databases do not scale well to deal with GWAS databases especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach which provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees may come at a serious price in terms of data utility. Building on such notions, we propose new methods to release aggregate GWAS data without compromising an individual’s privacy. We present methods for releasing differentially private minor allele frequencies, chi-square statistics and p-values. We compare these approaches on simulated data and on a GWAS study of canine hair length involving 685 dogs. We also propose a privacy-preserving method for finding genome-wide associations based on a differentially-private approach to penalized logistic regression.},
author = {Uhler, Caroline and Slavkovic, Aleksandra and Fienberg, Stephen},
journal = {Journal of Privacy and Confidentiality },
number = {1},
pages = {137 -- 166},
publisher = {Carnegie Mellon University},
title = {{Privacy-preserving data sharing for genome-wide association studies}},
doi = {10.29012/jpc.v5i1.629 },
volume = {5},
year = {2013},
}
@article{2959,
abstract = {We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.},
author = {Uhler, Caroline},
journal = {Annals of Statistics},
number = {1},
pages = {238 -- 261},
publisher = {Institute of Mathematical Statistics},
title = {{Geometry of maximum likelihood estimation in Gaussian graphical models}},
doi = {10.1214/11-AOS957},
volume = {40},
year = {2012},
}