The Geometry of Gaussoids
Tobias Boege, Alessio D'Ali, Thomas Kahle, and Bernd Sturmfels
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Submission date: 20. Oct. 2017
published in: Foundations of computational mathematics, 19 (2019) 4, p. 775-812
DOI number (of the published article): 10.1007/s10208-018-9396-x
Link to arXiv: See the arXiv entry of this preprint.
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 Matus 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.