Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear Rayleigh quotient optimization

Flavio Salizzoni, Luca Sodomaco, Julian Weigert

TL;DR

This work extends Rayleigh quotient optimization from quadratic forms to homogeneous polynomials of degree $\omega$ constrained to the affine cone over a projective variety $X$, introducing the X-eigenpoints and the Rayleigh-Ritz degree $\mathrm{RRdeg}_{\omega}(X)$. It establishes a deep link with Euclidean distance geometry by proving $\mathrm{RRdeg}_{\omega}(X)=\mathrm{DD}(\nu_{\omega}(X),Q_{BW})$ via the Bombieri-Weyl inner product and the Veronese embedding, enabling algebraic-geometric computation of critical points. The authors develop Porteous-type degeneracy formulas to compute or bound $\mathrm{RRdeg}_{\omega}(X)$ for implicit complete intersections and polynomial images, with explicit results for hypersurfaces and Segre-Veronese embeddings, complemented by Macaulay2 implementations. They also present a general-position theory that yields closed-form RR-degree expressions in terms of Chern classes and the generic-distance-degree framework, and a Segre-Veronese singular-tuple perspective that connects $X$-eigenvectors to singular $k$-tuples of multihomogeneous forms. The paper concludes with real-case considerations showing that RRdeg bounds may not tightly bound the number of real eigenpoints and provides concrete examples in rank-one tensor settings that illustrate the computational utility of the formulas.

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.

Nonlinear Rayleigh quotient optimization

TL;DR

This work extends Rayleigh quotient optimization from quadratic forms to homogeneous polynomials of degree constrained to the affine cone over a projective variety , introducing the X-eigenpoints and the Rayleigh-Ritz degree . It establishes a deep link with Euclidean distance geometry by proving via the Bombieri-Weyl inner product and the Veronese embedding, enabling algebraic-geometric computation of critical points. The authors develop Porteous-type degeneracy formulas to compute or bound for implicit complete intersections and polynomial images, with explicit results for hypersurfaces and Segre-Veronese embeddings, complemented by Macaulay2 implementations. They also present a general-position theory that yields closed-form RR-degree expressions in terms of Chern classes and the generic-distance-degree framework, and a Segre-Veronese singular-tuple perspective that connects -eigenvectors to singular -tuples of multihomogeneous forms. The paper concludes with real-case considerations showing that RRdeg bounds may not tightly bound the number of real eigenpoints and provides concrete examples in rank-one tensor settings that illustrate the computational utility of the formulas.

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 over a sphere, a projective algebraic variety , and we study the -eigenpoints of , which are classes of critical points of constrained to the sphere and the affine cone over . The number of -eigenpoints of a generic polynomial is the Rayleigh-Ritz degree of . This invariant is a version of the Euclidean distance degree of a Veronese embedding of . We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors.
Paper Structure (5 sections, 15 theorems, 91 equations, 3 figures)

This paper contains 5 sections, 15 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Rayleigh-Ritz degree of a projective variety
  3. Porteous-type formulas for Rayleigh-Ritz degrees
  4. Rayleigh-Ritz degrees of varieties in general position
  5. Rayleigh-Ritz degrees and singular tuples of tensors

Key Result

Lemma 2.5

Let $\langle\,,\rangle$ be an inner product on ${\mathbb R}^{n+1}$ and $\omega$ a positive integer. Let $\langle\,,\rangle_\mathsmaller{\mathrm{BW}}$ be the Bombieri-Weyl inner product on ${\mathbb R}[x]_{\omega}$ associated with $\langle\,,\rangle$. For any two polynomials $f=(f_\alpha)_{|\alpha|=\ we have the identities

Figures (3)

  • Figure 1: Comparison between normalized eigenvectors of $f$ (on the left) and critical points of the Bombieri-Weyl distance function from $f$ (on the right).
  • Figure 2: The output of HypersurfaceRegions.jl.
  • Figure 3: The real zero locus of the polynomial $g_3$ in \ref{['eq: polynomials g1 g2 g3']}.

Theorems & Definitions (42)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Example 2.8
  • ...and 32 more