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.

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Acknowledgements

We thank Moritz Firsching, Paul Görlach, Jon Hauenstein, Mateusz Michałek, Peter Nelson, Yue Ren, Caroline Uhler and Charles Wang for help with this project. Bernd Sturmfels was partially supported by the Einstein Foundation Berlin and the US National Science Foundation (DMS-1419018, DMS-1440140). Tobias Boege and Thomas Kahle were partially supported by the Deutsche Forschungsgemeinschaft (314838170, GRK 2297, “MathCoRe”).

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Correspondence to Thomas Kahle.

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Dedicated to the memory of František Matúš.

Communicated by Peter BUERGISSER.

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Cite this article

Boege, T., D’Alì, A., Kahle, T. et al. The Geometry of Gaussoids. Found Comput Math 19, 775–812 (2019). https://doi.org/10.1007/s10208-018-9396-x

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Keywords

  • Gaussoid
  • Matroid
  • Gaussian
  • Lagrangian Grassmannian
  • Minor
  • Symmetric matrix

Mathematics Subject Classification

  • 15A15
  • 60E05
  • 14M15
  • 13P10
  • 62H20
  • 17B10
  • 14T05