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Likelihood Geometry of Determinantal Point Processes

Hannah Friedman, Bernd Sturmfels, Maksym Zubkov

Abstract

We study determinantal point processes (DPP) through the lens of algebraic statistics. We count the critical points of the log-likelihood function, and we compute them for small models, thereby disproving a conjecture of Brunel, Moitra, Rigollet and Urschel.

Likelihood Geometry of Determinantal Point Processes

Abstract

We study determinantal point processes (DPP) through the lens of algebraic statistics. We count the critical points of the log-likelihood function, and we compute them for small models, thereby disproving a conjecture of Brunel, Moitra, Rigollet and Urschel.
Paper Structure (4 sections, 3 theorems, 33 equations)

This paper contains 4 sections, 3 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Three-by-three Matrices
  3. Partial Decouplings
  4. Numerical Computations

Key Result

Proposition 2.2

For $n=3$, the log-likelihood function in $\Theta$ given by some $u \in \mathcal{M}_3$ has critical points that do not correspond to partial decouplings. This resolves BMRU.

Theorems & Definitions (13)

  • Example 1.1: $n=2$
  • Example 2.1: Data in the model
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • Example 3.2
  • Example 3.3
  • proof : Proof of Theorem \ref{['thm:decoupling']}
  • Example 3.4
  • Proposition 4.1
  • ...and 3 more