Gaussian likelihood geometry of projective varieties

TL;DR

This paper investigates the maximum likelihood degree of homogeneous polynomials on projective varieties, linking it to geometric invariants like Euler characteristic, dual varieties, and Chern classes, thus generalizing Gaussian likelihood concepts.

Contribution

It introduces a geometric framework for understanding the maximum likelihood degree of polynomials on projective varieties, connecting it to various classical invariants.

Findings

Maximum likelihood degree equals the count of critical points of a rational function on the variety.

Provides geometric characterizations of the maximum likelihood degree via Euler characteristic, dual varieties, and Chern classes.

Generalizes the Gaussian maximum likelihood degree to broader algebraic geometric contexts.

Abstract

We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_F(X)$ is equal to the count of critical points of a rational function on $X$, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.