Support Sets in Exponential Families and Oriented Matroid Theory

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.Comment: 27 pages, extended version published in IJA

Constructions and complexity of secondary polytopes

AbstractThe secondary polytope Σ(A) of a configuration A of n points in affine (d − 1)-space is an (n − d)-polytope whose vertices correspond to regular triangulations of conv(A). In this article we present three constructions of Σ(A) and apply them to study various geometric, combinatorial, and computational properties of secondary polytopes. The first construction is due to Gel'fand, Kapranov, and Zelevinsky, who used it to describe the face lattice of Σ(A). We introduce the universal polytope u(A) ⊂ ΛdRn, a combinatorial object depending only on the oriented matroid of A. The secondary Σ(A) can be obtained as the image of u(A) under a canonical linear map onto Rn. The third construction is based upon Gale transforms or oriented matroid duality. It is used to analyze the complexity of computing Σ(A) and to give bounds in terms of n and d for the number of faces of Σ(A)

Diameter and Coherence of Monotone Path Graph

University of Minnesota Ph.D. dissertation. May 2015. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); ix, 93 pages.A Zonotope $Z$ is the linear projection of an $n$-cube into $\mathbb{R}^d$. Given a generic linear function $f$, an $f$-monotone path on $Z$ is a path along edges from the $f$-minimizing vertex $-z$ to its opposite vertex $z$. The monotone paths of $Z$ are the vertices of the monotone path graph in which two $f$-monotone paths are adjacent when they differ in a face of $Z$. In our illustration the two red paths are adjacent in the monotone path graph because they differ in the highlighted face. An $f$-monotone path is coherent if it lies on the boundary of a polygon obtained by projecting $Z$ to 2 dimensions. The dotted, thick, red path in Figure 0.1 (see pdf) is coherent because it lies on the boundary after projecting $Z$ to the page. However, there is no equivalent projection for the blue double path. The alternate red path may be coherent or incoherent based on the choice of $f$. The coherent $f$-monotone paths of $Z$ are a set of geometrically distinguished galleries of the monotone path graph. Classifying when incoherent $f$-monotone paths exist is the central question of this thesis. We provide a complete classification of all monotone path graphs in corank 1 and 2, finding all families in which every $f$-monotone path is coherent and showing that all other zonotopes contain at least one incoherent $f$-monotone path. For arrangements of corank 1, we prove that the monotone path graph has diameter equal to the lower bound suggested by Reiner and Roichman using methods of $L_2$-accessibility and illustrate that $L_2$ methods cannot work in corank 2 by finding a monotone path graph which has no $L_2$-accessible nodes. We provide examples to illustrate the monotone path graph and obtain a variety of computational results, of which some are new while others confirm results obtained through different methods. Our primary methods use duality to reformulate coherence as a system of linear inequalities. We classify monotone path graphs using single element liftings and extensions, proving for when $Z$ has incoherent $f$-monotone paths, then any lifting or extension of $Z$ has incoherent $f$-monotone paths too. We complete our classification by finding all monotone path graphs with only coherent $f$-monotone paths and finding a set of minimal obstructions which always have incoherent $f$-monotone paths

On Boundaries of Statistical Models

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics