@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{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{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{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{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}, } @misc{2007, abstract = {Maximum likelihood estimation under relational models, with or without the overall effect. For more information see the reference manual}, author = {Klimova, Anna and Rudas, Tamás}, publisher = {The Comprehensive R Archive Network}, title = {{gIPFrm: Generalized iterative proportional fitting for relational models}}, year = {2014}, } @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}, } @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{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}, } @unpublished{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}, booktitle = {arXiv}, title = {{Sphere packing with limited overlap}}, doi = {10.48550/arXiv.1401.0468}, year = {2014}, }