Maximum Likelihood, permutohedra and Associativity Equations

TL;DR

This paper explores the geometric and algebraic structures of concentration matrix cones, showing their relation to Monge-Ampère domains, associativity equations, and permutohedra, with implications for maximum likelihood estimation.

Contribution

It establishes that the cone of concentration matrices is a Monge-Ampère domain and links the moduli space to associativity equations and permutohedra, revealing new geometric insights.

Findings

The cone of concentration matrices is a Monge-Ampère domain.

The moduli space satisfies the Associativity Equations.

Maximum Likelihood degree is indexed by Frobenius residuals.

Abstract

We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge--Amp\`ere 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 Bia\l{}ynicki--Birula cells. We prove that the Maximum Likelihood degree is indexed by components lying on those Frobenius residuals.