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Likelihood Geometry of the Squared Grassmannian

Hannah Friedman

TL;DR

The paper connects projection DPPs to the squared Grassmannian and computes the likelihood geometry of the d=2 case. Using topological methods on open very affine varieties and a stratified fibration framework, it proves the ML degree of ${\rm sGr}(2,n)$ is $\frac{(n-1)!}{2}$ and that all critical points are real and nonnegative, with all being local maxima on the real locus; numerical experiments for higher d (e.g., $d=3,n=6$) suggest a rich real solution structure and provide lower bounds via real sign patterns. The results advance the algebraic statistics of DPPs, linking Plücker coordinates, Euler characteristics, and likelihood optimization in a complete treatment for the $d=2$ projection case, and outlining challenges and observations for higher dimensions.

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.

Likelihood Geometry of the Squared Grassmannian

TL;DR

The paper connects projection DPPs to the squared Grassmannian and computes the likelihood geometry of the d=2 case. Using topological methods on open very affine varieties and a stratified fibration framework, it proves the ML degree of is and that all critical points are real and nonnegative, with all being local maxima on the real locus; numerical experiments for higher d (e.g., ) suggest a rich real solution structure and provide lower bounds via real sign patterns. The results advance the algebraic statistics of DPPs, linking Plücker coordinates, Euler characteristics, and likelihood optimization in a complete treatment for the projection case, and outlining challenges and observations for higher dimensions.

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 critical points, all of which are real and positive, thereby settling a conjecture of Devriendt, Friedman, Reinke, and Sturmfels.
Paper Structure (4 sections, 13 theorems, 31 equations, 1 figure)

This paper contains 4 sections, 13 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Parametric Model
  3. Stratification of the Parametric Model
  4. Real Solutions

Key Result

Lemma 1.1

The principal minors of an orthogonal projection matrix $P$ are proportional to the corresponding squared Plücker coordinates:

Figures (1)

  • Figure 1: Posets for the special strata. The values of the Möbius function $\mu(-, S_i)$ and $\mu(-, S_i \cap S_j)$, respectively, are shown in blue.

Theorems & Definitions (24)

  • Lemma 1.1: DFRS, Lemma 3.1
  • Theorem 1.2: DFRS, Conjecture 4.2
  • Theorem 1.3
  • Example 1.4: $d = 2, n = 3$
  • Theorem 2.1: huh, Theorem 1
  • Lemma 2.2
  • proof
  • Lemma 3.1: ABFKSTL, Lemma 2.3
  • Lemma 3.2
  • proof
  • ...and 14 more