The maximum likelihood degree of Fermat hypersurfaces
Agostini, Daniele
Creator
Faculty
paolo.lella@polimi.it
Alberelli, Davide
Creator
Grande, Francesco
Creator
Lella, Paolo
Creator
MATH / Applied Mathematics
Illinois Institute of Technology
Affiliated department
Maximum likelihood
Fermat hypersurface
Likelihood correspondence
2015
2015-11-09
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.
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