Maximum Likelihood Duality for Determinantal Varieties

TL;DR

This paper proves a conjecture relating critical points of likelihood functions on different matrix varieties, establishing a duality for rectangular, symmetric, and skew-symmetric matrices, advancing algebraic statistics and matrix theory.

Contribution

It confirms a conjecture about likelihood duality for various matrix varieties, providing rigorous proofs for rectangular, symmetric, and skew-symmetric cases.

Findings

Proved likelihood duality for rectangular matrices.

Established duality for symmetric matrices.

Extended results to skew-symmetric matrices.

Abstract

In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of co-rank r-1. In this paper, we prove that conjecture for rectangular matrices and for symmetric matrices, as well as a variant for skew-symmetric matrices. To appear in International Mathematics Research Notices.