The Geometry of Gaussoids

  • Tobias Boege
  • Alessio D’Alì
  • Thomas KahleEmail author
  • Bernd Sturmfels


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.


Gaussoid Matroid Gaussian Lagrangian Grassmannian Minor Symmetric matrix 

Mathematics Subject Classification

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



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”).


  1. 1.
    F. Ardila, F. Rincón and L. Williams: Positively oriented matroids are realizable, J. Eur. Math. Soc. (JEMS) 19 (2017) 815–833.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Baker and N. Bowler: Matroids over partial hyperstructures, arXiv:1709.09707.
  3. 3.
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler: Oriented Matroids, Cambridge University Press, 1993.Google Scholar
  4. 4.
    J. Bokowski and J. Richter: On the finding of final polynomials, European J. Combinatorics 11 (1990) 21–34.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Borovik, I.M. Gelfand and N. White: Coxeter Matroids, Progress in Mathematics 216, Birkhäuser, Boston, MA, 2003.CrossRefzbMATHGoogle Scholar
  6. 6.
    S. Brodsky, C. Ceballos and J-P. Labbé: Cluster algebras of type D4, tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom. 58 (2017) 25–46.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Dress and W. Wenzel: Valuated matroids: a new look at the greedy algorithm, Appl. Math. Letters 3 (1990) 33–35.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Dress and W. Wenzel: Grassmann–Plücker relations and matroids with coefficients, Adv. Math. 86 (1991) 68–110.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Drton, B. Sturmfels and S. Sullivant: Lectures on Algebraic Statistics, Oberwolfach Seminars, 39, Birkhäuser Verlag, Basel, 2009.CrossRefzbMATHGoogle Scholar
  10. 10.
    M. Drton and H. Xiao: Smoothness of Gaussian conditional independence models, Algebraic methods in statistics and probability II, 155–177, Contemporary Mathematics, 516, Amer. Math. Soc., Providence, RI, 2010.CrossRefzbMATHGoogle Scholar
  11. 11.
    S. Fallat, S. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth and P. Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) 1152–1184.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    K. Fukuda, H. Miyata and S. Moriyama: Complete enumeration of small realizable oriented matroids, Discrete Comput. Geom. 49 (2013) 359–381.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    W. Fulton and J. Harris: Representation Theory, Graduate Texts in Mathematics 129, Springer, New York, 1991.zbMATHGoogle Scholar
  14. 14.
    P. Görlach, Y. Ren and J. Sommars: Detecting tropical defects of polynomial equations, in preparation.Google Scholar
  15. 15.
    D. Grayson and M. Stillman: Macaulay2, a software system for research in algebraic geometry, available at
  16. 16.
    H. Hiller: Combinatorics and intersection of Schubert varieties, Comment. Math. Helv. 57 (1982) 41–59.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O. Holtz and B. Sturmfels: Hyperdeterminantal relations among symmetric principal minors, Journal of Algebra 316 (2007) 634–648.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Iliev and K. Ranestad: Geometry of the Lagrangian Grassmannian \(LG(3,6)\) with applications to Brill–Noether loci, Michigan Math. J. 53(2) (2005) 383–417.Google Scholar
  19. 19.
    H. Joe: Generating random correlation matrices based on partial correlations, J. Multivariate Analysis 97 (2006) 2177–2189.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. Karlin and Y. Rinott: M-matrices as covariance matrices of multinormal distributions, Linear Algebra Appl. 52 (1983) 419–438.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    K. Kaveh and C. Manon: Khovanskii bases, higher rank valuations and tropical geometry, arXiv:1610.00298.
  22. 22.
    R. Kenyon and R. Pemantle: Principal minors and rhombus tilings, J. Phys. A 47 (2014) 474010, 17 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    S. Lauritzen, C. Uhler and P. Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, Annals of Statistics, to appear.Google Scholar
  24. 24.
    R. Lněnička and F. Matúš: On Gaussian conditional independence structures, Kybernetika 43 (2007) 327–342.MathSciNetzbMATHGoogle Scholar
  25. 25.
    D. Maclagan and B. Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, Providence, RI, 2015.CrossRefzbMATHGoogle Scholar
  26. 26.
    S. J. Maher et al.: The SCIP Optimization Suite 4.0, Zuse Institute, Berlin.Google Scholar
  27. 27.
    D. Mayhew, G. Whittle and M. Newman: Is the missing axiom of matroid theory lost forever? The Quarterly Journal of Mathematics 65(4) (2014) 1397–1415.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    D. Mayhew, M. Newman and G. Whittle: Yes, the “missing axiom” of matroid theory is lost forever, Transactions of the American Mathematical Society, to appear (2018)Google Scholar
  29. 29.
    F. Mohammadi, C. Uhler, C. Wang and J. Yu: Generalized permutohedra from probabilistic graphical models, SIAM Journal on Discrete Mathematics 32 (2018) 64–93.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    L. Oeding: G-Varieties and the Principal Minors of Symmetric Matrices, PhD Dissertation, Texas A&M University, ProQuest LLC, Ann Arbor, MI, 2009.Google Scholar
  31. 31.
    L. Oeding: Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebra and Number Theory 5 (2011) 75–109.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    K. Sadeghi: Faithfulness of probability distributions and graphs, Journal of Machine Learning Research 18 (2017) 1–29.MathSciNetzbMATHGoogle Scholar
  33. 33.
    SageMath, the Sage Mathematics Software System (Version 8.0), The Sage Developers, 2017,
  34. 34.
    P. Šimeček: Gaussian representation of independence models over four random variables In: Proc. COMPSTAT 2006, World Conference on Computational Statistics 17 (A. Rizzi and M. Vichi, eds.), Rome 2006, pp. 1405–1412.Google Scholar
  35. 35.
    D. Speyer and L. Williams: The tropical totally positive Grassmannian, J. Algebraic Combinatorics 22 (2005) 189–210.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. Studeny: Probabilistic Conditional Independence Structures, Information Science and Statistics, Springer, London, 2005.zbMATHGoogle Scholar
  37. 37.
    B. Sturmfels: Open problems in algebraic statistics, Emerging Applications of Algebraic Geometry. edited by M. Putinar and S. Sullivant, Springer, New York (2009), 351–363.CrossRefGoogle Scholar
  38. 38.
    B. Sturmfels, E. Tsukerman and L. Williams: Symmetric matrices, Catalan paths, and correlations, J. Combinatorial Theory, Ser. A 144 (2016) 496–510.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    S. Sullivant: Gaussian conditional independence relations have no finite complete characterization, Journal of Pure and Applied Algebra 213 (2009) 1502–1506.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    M. Thurley: sharpSAT - Counting models with advanced component caching and implicit BCP, Proc. 9th Int. Conf. Theory and Applications of Satisfiability Testing (SAT 2006), (2006) pp. 424–429.Google Scholar
  41. 41.
    T. Toda and S. Takehide: Implementing efficient all solutions SAT solvers, J. Experimental Algorithmics 21.1 (2016) 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    P. Vámos: The missing axiom of matroid theory is lost forever, Journal of the London Mathematical Society 18 (1978) 403–408.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2018

Authors and Affiliations

  • Tobias Boege
    • 1
  • Alessio D’Alì
    • 2
    • 4
  • Thomas Kahle
    • 1
    Email author
  • Bernd Sturmfels
    • 2
    • 3
  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany
  2. 2.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.University of California BerkeleyBerkeleyUSA
  4. 4.Mathematics InstituteUniversity of WarwickCoventry CV4 7ALUK

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