Gaussian graphical models with toric vanishing ideals
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Gaussian graphical models with toric vanishing ideals Pratik Misra1 · Seth Sullivant1 Received: 16 December 2019 / Revised: 2 September 2020 / Accepted: 4 September 2020 © The Institute of Statistical Mathematics, Tokyo 2020
Abstract Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures given by Sturmfels and Uhler. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree 1 and 2. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the 1-clique sums of complete graphs. Keywords Clique sum · Toric ideals · SAGBI bases · Initial algebra
1 Introduction Any positive definite n × n matrix 𝛴 can be seen as the covariance matrix of a multivariate normal distribution in ℝn . The inverse matrix K = 𝛴 −1 is called the concentration matrix of the distribution, which is also positive definite. The statistical models where the concentration matrix K can be written as a linear combination of some fixed linearly independent symmetric matrices K1 , K2 , … , Kd are called linear concentration models. Let 𝕊n denote the vector space of real symmetric matrices and let L be a linear subspace of 𝕊n generated by K1 , K2 , … , Kd . The set L−1 is defined as
L−1 = {𝛴 ∈ 𝕊n ∶ 𝛴 −1 ∈ L}.
* Pratik Misra [email protected] Seth Sullivant [email protected] 1
Department of Mathematics, North Carolina State University, 2108 SAS Hall Box 8205, Raleigh, NC 27695, USA
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The homogeneous ideal of all the polynomials in ℝ[𝛴] = ℝ[𝜎11 , 𝜎12 , … , 𝜎nn ] that vanish on L−1 is denoted by PL . Note that PL is prime because it is the vanishing ideal of L−1 , which is the image of the irreducible variety L under the rational inversion map. In this paper, we study the problem of finding a generating set of PL for the special case of Gaussian graphical models. Gaussian graphical models are used throughout the natural sciences and especially in computational biology as seen in Koller and Friedman (2009) and Lauritzen (1996). These models explicitly capture the statistical relationships between the variables of interest in the form of a graph. The undirected Gaussian graphical model is obtained when the subspace L of 𝕊n is defined by the vanishing of some off-diagonal entries of the concentration matrix K. We fix a graph G = ([n], E) with vertex set [n] = {1, 2, … , n} and edge set E, which is assumed to contain all self loops. The subspace L is generated by the set {Kij |(i, j) ∈ E} of matrices Kij with 1 entry at the (i, j)th and (j, i)th position and 0 in all other positions. We denote the ideal PL as PG in this model. One way to compute PG is to eliminate the entries of an indeterminate symmetric n × n m
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