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 Lnenicka and Matús 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.

32 pages, 4 figures, v2: Prop. 6.4 and Thm. 8.4 added, various small improvements, 34 pages

References in corpus (2)

• Hyperdeterminantal relations among symmetric principal minors
• Implementing Efficient All Solutions SAT Solvers

Cited by in corpus (7)

• Gaussoids are two-antecedental approximations of Gaussian conditional independence structures
• The geometry of Gaussian double Markovian distributions
• Construction Methods for Gaussoids
• Detecting tropical defects of polynomial equations
• The quasi principal rank characteristic sequence
• Selfadhesivity in Gaussian conditional independence structures
• Self-adhesivity in lattices of abstract conditional independence models