Exponential Varieties

TL;DR

This paper develops a comprehensive theory of exponential varieties, connecting hyperbolic polynomials, algebraic geometry, and statistical models like Gaussian graphical models, highlighting their positivity, convexity, and algebraic properties.

Contribution

It introduces a general framework for exponential varieties derived from hyperbolic polynomials, linking algebraic, geometric, and statistical aspects.

Findings

Comparison of multidegrees and ML degrees of gradient maps

Identification of exponential varieties with inverse symmetric matrices

Connection between hyperbolic polynomials and algebraic statistical models

Abstract

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials.