Nonlinear Rayleigh quotient optimization

TL;DR

This paper extends Rayleigh quotient optimization from quadratic forms to homogeneous polynomial functions over algebraic varieties, introducing the concept of $X$-eigenpoints and relating them to algebraic invariants like the Rayleigh-Ritz degree.

Contribution

It introduces the notion of $X$-eigenpoints for polynomial optimization over varieties and connects this to algebraic invariants such as the Rayleigh-Ritz degree, providing formulas for various cases.

Findings

Defined $X$-eigenpoints as critical points of polynomial functions constrained to varieties.

Connected the number of $X$-eigenpoints to the Rayleigh-Ritz degree and Euclidean distance degree.

Provided explicit formulas for $X$-eigenpoints in scenarios involving rank-one tensors.

Abstract

Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function $f$ over a sphere, a projective algebraic variety $X$, and we study the $X$-eigenpoints of $f$, which are classes of critical points of $f$ constrained to the sphere and the affine cone over $X$. The number of $X$-eigenpoints of a generic polynomial $f$ is the Rayleigh-Ritz degree of $X$. This invariant is a version of the Euclidean distance degree of a Veronese embedding of $X$. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors.