The Geometry of Gaussoids

Abstract

A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lněnička and Matúš are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models. a 12|3 maps to σ 12 σ 33 − σ 13 σ 23 whereas a 13|2 maps to −(σ 12 σ 23 − σ 13 σ 22 ). Maintaining this sign convention is important to keep the algebra consistent with its statistical interpretation.Let J n denote the ideal generated by all homogeneous polynomials in the kernel of the map above. This defines an irreducible variety V (J n ) of dimension n+1 2 in the projective space P 2 n−2 (4+( n 2 ))−1 whose coordinates are P ∪ A. There is a natural projection from LGr(n, 2n) onto V (J n ), obtained by deleting all minors that are neither principal nor almost-principal. This is analogous to [30, Observation III.12], where the focus was on principal minors p I .Proposition 1.1. The degree of the projective variety of principal and almost-principal minors coincides with the degree of the Lagrangian Grassmannian. For n ≥ 2, it equals degree(V (J n )) = degree(LGr(n, 2n)) = n+1 2 ! 1 n · 3 n−1 · 5 n−2 · · · (2n − 1).

Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

“…This group is, in turn, a discrete subgroup of the SL 2 (R) N action on the Lagrangian Grassmannian; cf. [3,14]. In the semidirect product, every group element can be written as the composition of an element of S N and one of (Z/4) N .…”

Detecting tropical defects of polynomial equations

“…◮ a tropical Grassmannian inside its Dressian [HJS14], ◮ matrices of Kapranov rank r as a subset of matrices of tropical rank r [DSS05], ◮ the tropicalization of total coordinate spaces of smooth cubic surfaces inside the prevariety cut out by the relations arising from isotropic planes [RSS16], ◮ the realizability locus inside the space of gaussoids [BDKS17], ◮ the realizability locus inside the space of all ∆-matroids [Rin12].…”

“…In Section 3, we utilize the algorithms to disprove Conjecture 5.3 in [RSS16] (Theorem 3.2), which is an algebraic hint of the known discrepancy between lines on a tropical cubic and lines on an algebraic cubic [Vig10]. In Section 4, we employ the algorithms to disprove Conjecture 8.4 in [BDKS17], which shows that there are non-realizable gaussoids of rank 4. Our attempts on Conjecture 4.8 in [Rin12] were inconclusive.…”

“…and Matúš[LM07] that encode conditional independence relations among Gaussian random variables. Reminiscent to the study of matroids, Boege, D'Alì, Kahle and Sturmfels[BDKS17] studied them algebraically, introducing the notions of oriented and valuated gaussoids. In this section, we will address their conjecture on whether all valuated gaussoids over a four-element set are realizable.…”

The quasi principal rank characteristic sequence

“…For example, almost-principal minors (and principal minors) are of interest in algebraic geometry and statistics: In [14], Sturmfels presented mathematical problems whose solutions would, according to him, "likely become important contributions to the emerging interactions between algebraic geometry and computational statistics." One of the problems he presented is to study the geometry of conditional independence models for multivariate Gaussian random variables, where almost-principal minors play a crucial role, due to their intimate connection with conditional independence in statistics [14]; the program presented in [14] was recently continued by Boege, D'Alì, Kahle and Sturmfels in [2]. We note that almost-principal minors are referred to as partial covariance (or, if renormalized, partial correlations) in the statistics literature [14].…”