Likelihood Geometry of the Squared Grassmannian

TL;DR

This paper investigates the likelihood geometry of the squared Grassmannian, proving that the log-likelihood function has a specific number of real, positive critical points, confirming a conjecture in algebraic statistics.

Contribution

It establishes the exact count and nature of critical points of the likelihood function for projection determinantal point processes on the squared Grassmannian, settling a prior conjecture.

Findings

The log-likelihood function has (n - 1)!/2 critical points.

All critical points are real and positive.

The result confirms a conjecture by Devriendt et al.

Abstract

We study projection determinantal point processes and their connection to the squared Grassmannian. We prove that the log-likelihood function of this statistical model has $(n - 1)!/2$ critical points, all of which are real and positive, thereby settling a conjecture of Devriendt, Friedman, Reinke, and Sturmfels.