Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMatroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models
108
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with hidden random variables corresponds to a variety defined by certain determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We show that any CI variety, and more generally any hypergraph variety, admits a matroid stratification. Our main motivation for studying decompositions of CI varieties is the realizability problem: given a collection of CI relations, the goal is to determine the existence of random variables that satisfy these constraints and violates the rest. We show that the realization spaces of CI models and the matroid varieties in their decompositions are closely related. We use ideas from incidence geometry, in particular point and line configurations, to find minimal decompositions of general hypergraph varieties in terms of matroid varieties, which are not necessarily irreducible by Mnev--Sturmfels universality theorem, and may have arbitrary singularities. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. Our decomposition strategy gives immediate matroid interpretations of the irreducible components of many families of CI varieties in algebraic statistics, and unravels the symmetric structures in CI varieties which hugely simplifies the computations.
rate research
Read More
Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We aim to demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the LCI models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series, and Shannon information flow.
This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. This is accomplished by computing primary decompositions of conditional independence ideals. As examples the chapter presents in detail the graphical model of a four cycle and the intersection axiom, a certain implication of conditional independence statements. Another important problem in the area is to determine all constraints on a graphical model, for example, equations determined by trek separation. The full set of equality constraints can be determined by computing the models vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this varietys cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springers work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikovs theory of generalized permutohedra.
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from $K_4$ graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Groebner Basis calculation took 5 days and 6 hrs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
