The maximum likelihood degree of Fermat hypersurfaces

TL;DR

This paper investigates the maximum likelihood degree of Fermat hypersurfaces, providing explicit formulas, algorithms exploiting symmetries, and computational results for various cases, advancing understanding in algebraic statistics.

Contribution

It offers closed-form formulas for the ML degree of Fermat curves and hypersurfaces, along with symmetry-based algorithms for computation and extensive numerical analysis.

Findings

Closed formulas for Fermat curves and degree 2 hypersurfaces

Algorithms leveraging symmetries for ML degree computation

Computational results for multiple Fermat hypersurfaces

Abstract

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.