Exponential Varieties
Mateusz Micha{\l}ek, Bernd Sturmfels, Caroline Uhler, Piotr Zwiernik

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
