Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

TL;DR

This paper explores deep connections between algebraic geometry, optimization, and statistics, establishing new formulas and confirming conjectures about the degrees of certain models and programs, with implications for understanding their asymptotic behavior.

Contribution

It proves conjectures relating ML-degree, SDP algebraic degree, and Schubert calculus, providing explicit formulas and extending results to broader matrix spaces.

Findings

Proved polynomiality of the ML-degree for linear concentration models.

Derived explicit formulas for the algebraic degree of semidefinite programming.

Extended results to spaces of general and skew-symmetric matrices.

Abstract

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.