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Maximum Likelihood, permutohedra and Associativity Equations

Noémie C. Combe

TL;DR

The work develops a unified geometric framework for the cone of concentration matrices arising in linear concentration models and Wishart laws, showing that the interior cone is a Hessian/Kähler Monge–Ampère domain and that its tangent bundle carries a pre-Lie algebra. It then connects the diagonal spectrahedron to the Associativity Equations, embedding this structure into Frobenius-manifold/mirror-symmetry context and revealing a toric permutohedral compactification related to Losev–Manin spaces. The boundary analysis via BB cells identifies Frobenius residuals that parametrize the compactified space and align with permutohedron geometry, providing a combinatorial and toric description of the Frobenius/ML structure. Finally, the ML degree is interpreted through the geometry of complete quadrics, expressed as a sum over torus-fixed points on toric/complete-quadric strata, linking algebraic statistics with mirror symmetry and toric degenerations.

Abstract

We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge--Ampère domain and that the log-likelihood function generates its potential function at the identity. The tangent sheaf carries the structure of a pre-Lie algebra. We also show that the moduli space of diagonal matrices parameterizing the polyhedral spectrahedron satisfies the Associativity Equations, a notion central in mirror symmetry, and that its compactification is a toric variety associated to a permutohedron, reminiscent to Losev--Manin spaces. Finally we introduce Frobenius residuals: these are connected components of the compactified Frobenius manifold of diagonal matrices, generated by the Białynicki--Birula cells. We prove that the Maximum Likelihood degree is indexed by components lying on those Frobenius residuals.

Maximum Likelihood, permutohedra and Associativity Equations

TL;DR

The work develops a unified geometric framework for the cone of concentration matrices arising in linear concentration models and Wishart laws, showing that the interior cone is a Hessian/Kähler Monge–Ampère domain and that its tangent bundle carries a pre-Lie algebra. It then connects the diagonal spectrahedron to the Associativity Equations, embedding this structure into Frobenius-manifold/mirror-symmetry context and revealing a toric permutohedral compactification related to Losev–Manin spaces. The boundary analysis via BB cells identifies Frobenius residuals that parametrize the compactified space and align with permutohedron geometry, providing a combinatorial and toric description of the Frobenius/ML structure. Finally, the ML degree is interpreted through the geometry of complete quadrics, expressed as a sum over torus-fixed points on toric/complete-quadric strata, linking algebraic statistics with mirror symmetry and toric degenerations.

Abstract

We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge--Ampère domain and that the log-likelihood function generates its potential function at the identity. The tangent sheaf carries the structure of a pre-Lie algebra. We also show that the moduli space of diagonal matrices parameterizing the polyhedral spectrahedron satisfies the Associativity Equations, a notion central in mirror symmetry, and that its compactification is a toric variety associated to a permutohedron, reminiscent to Losev--Manin spaces. Finally we introduce Frobenius residuals: these are connected components of the compactified Frobenius manifold of diagonal matrices, generated by the Białynicki--Birula cells. We prove that the Maximum Likelihood degree is indexed by components lying on those Frobenius residuals.
Paper Structure (19 sections, 11 theorems, 37 equations)

This paper contains 19 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Setting: a semi-algebraic cone of matrices
  3. Cone of concentration matrices and sufficient statistics
  4. Spectrahedron
  5. Associativity Equations, permutohedra and BB cells
  6. Geometry of convex cones, spectrahedra and their properties
  7. Affine and Hessian structures
  8. Affine structures
  9. Pre-Lie structures of the tangent sheaf
  10. Characteristic function for $\mathscr{S}^n_{\geq 0}$
  11. Hessian structures
  12. The cone $\mathscr{S}^n_{>0}$ is a Monge--Ampère domain
  13. Log-likelihood and Maximum--Likelihood
  14. Cone of concentration matrices
  15. Complete quadrics, log-likelihood and ML degree
  16. ...and 4 more sections

Key Result

Theorem 1

The open cone $\mathscr{S}^n_{>0}$ is an affine domain endowed with a Hessian structure and the tangent sheaf to $\mathscr{S}^n_{>0}$ carries the structure of a pre-Lie algebra.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 14 more