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Decomposing Determinantal Varieties from Statistics via Matroid Theory

Per Alexandersson, Yulia Alexandr, Emiliano Liwski, Fatemeh Mohammadi, Pardis Semnani

TL;DR

This work integrates algebraic statistics and matroid theory to study determinantal varieties arising from conditional independence models with hidden variables. It develops a systematic decomposition framework: V_Δ is expressed as a union of auxiliary varieties V_S, whose irreducible components are controlled by quasi-affine U_S and lower-dimensional FS, with a precise one-to-one correspondence between components of F_S and U_S. The authors obtain explicit decompositions and dimension formulas, especially for the case k=2 and for t=ℓ, and provide defining equations for the components via transfers from F_S. A highlight is the degree computation, reduced to lattice-path enumeration and a combinatorial hypergraph transversals problem, yielding exact formulas and generating functions; this connects to quasi-products of matroids and broadens the toolkit for algebraic statistics beyond current computational methods.

Abstract

We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gröbner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.

Decomposing Determinantal Varieties from Statistics via Matroid Theory

TL;DR

This work integrates algebraic statistics and matroid theory to study determinantal varieties arising from conditional independence models with hidden variables. It develops a systematic decomposition framework: V_Δ is expressed as a union of auxiliary varieties V_S, whose irreducible components are controlled by quasi-affine U_S and lower-dimensional FS, with a precise one-to-one correspondence between components of F_S and U_S. The authors obtain explicit decompositions and dimension formulas, especially for the case k=2 and for t=ℓ, and provide defining equations for the components via transfers from F_S. A highlight is the degree computation, reduced to lattice-path enumeration and a combinatorial hypergraph transversals problem, yielding exact formulas and generating functions; this connects to quasi-products of matroids and broadens the toolkit for algebraic statistics beyond current computational methods.

Abstract

We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gröbner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.
Paper Structure (30 sections, 49 theorems, 144 equations, 2 figures, 1 table)

This paper contains 30 sections, 49 theorems, 144 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation.
  3. From conditional independence to determinantal ideals.
  4. The model.
  5. Definitions
  6. Algebraic formulation
  7. Matroid definitions
  8. Contributions
  9. A matroid-theoretic approach.
  10. Outline of the strategy.
  11. General decomposition theorem
  12. Decomposition into varieties VS
  13. Decomposition into varieties FS
  14. Dimension of the components of $V_{\Delta}$ for arbitrary $k$
  15. Example: Irreducible decomposition of VDelta for t=l
  16. ...and 15 more sections

Key Result

Theorem (A)

The variety $V_{\Delta}$ admits the following irredundant irreducible decomposition: where the union runs over all admissible subsets $S \subset [k\ell]$ such that $V_{S} \neq \emptyset$, and where $V_{S,i}$ denote the irreducible components of $V_{S}$. Moreover:

Figures (2)

  • Figure 1: Illustration of the perturbation of the vector $v_{3}$ from Example \ref{['example perturbation']}.
  • Figure 2: (Left) Two families of WS non-intersecting paths from Remark \ref{['rem:two_paths']}. (Right) Three lattice paths in the case $\ell=9$, $d=7$ and $t-1=3$. The path $P_1$ has edge weight $z_1^3 z_3 z_5^2$, $P_2$ has weight $z_2^2 z_5 z_6 z_7$ and finally $P_3$ has weight $z_4 z_7^2 z_8$.

Theorems & Definitions (132)

  • Definition 1.1: CI ideal
  • Definition 1.2
  • Definition 1.3
  • Theorem (A)
  • Theorem (B)
  • Theorem (C)
  • Corollary 1.4
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 122 more