Squared Linear Models

TL;DR

Squared linear models parametrize probabilities by $p_i(x) = \ell_i^2(x)/\sum_j \ell_j^2(x)$, yielding a rich likelihood geometry. The paper proves that all complex critical points of the log-likelihood are real and positive, with one per region of the projective hyperplane arrangement, and provides a determinantal presentation of the likelihood correspondence, along with tropical degenerations and log-normal polytopes. It further links these models to Veronese projections and discriminantal/linear projection determinantal point processes, offering explicit formulas for ML degree and showcasing examples such as braid and Steiner arrangements. Collectively, the results illuminate the global optimization landscape for squared linear models and equip practitioners with exact, algebraic tools for inference and combinatorial analysis.

Abstract

We study statistical models that are parametrized by squares of linear forms. All critical points of the likelihood function are real and positive. There is one critical point in each region of the projective hyperplane arrangement defined by the linear forms. We examine the ideal and singular locus of the model, and we give a determinantal presentation for its likelihood correspondence. We characterize tropical degenerations of the MLE, we describe the log-normal polytopes, and we explore connections to determinantal point processes.

Figures (3)

• Figure 1: Tropical MLE for $d=3,n=4$ gives a bijection between the seven regions of $\mathbb{P}^2_\mathbb{R} \backslash \mathcal{A}$ and the seven faces of the triangle. Each arc $y(\epsilon)$ travels from its region to the triangle.
• Figure 1: The squared linear model (blue) shown inside the triangle $\Delta_2$ of data (black), together with its logarithmic normal bundle (gray dashed lines). The triangle is divided into six Weyl chambers (red lines). The log-Voronoi cell of the point $p$ is the intersection of the fiber of the logarithmic normal bundle with the corresponding Weyl chamber (green line).
• Figure 2: The log-normal polygon in Example \ref{['ex:nongeneric-uniform2']}. The log-Voronoi cell is marked in green.

Theorems & Definitions (45)

• Example 1.1: $d=3,n=4$
• Example 1.2: The braid arrangement
• Theorem 2.1
• Lemma 2.2
• Proof 1
• Remark 2.3
• Proof 2: Proof of Theorem \ref{['thm:two']}
• Corollary 2.4
• Example 2.5
• Remark 2.6