The Geometry of Gaussoids

• Published in: Foundations of computational mathematics Vol. 19; no. 4; pp. 775 - 812
• Main Authors: Boege, Tobias, D’Alì, Alessio, Kahle, Thomas, Sturmfels, Bernd
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-08-2019; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Axioms; Combinatorial analysis; Computer Science; Economics; Lagrangian functions; Linear algebra; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical research; Mathematics; Mathematics and Statistics; Matrices; Matrix methods; Matrix Theory; Numerical Analysis; Germany; 14M15; Matroid; 13P10; 15A15; Gaussoid; Lagrangian Grassmannian; Symmetric matrix; 17B10; Gaussian; 62H20; Minor; 14T05; 60E05
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-018-9396-x

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.