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Distance Optimization in the Grassmannian of Lines

Hannah Friedman, Andrea Rosana, Bernd Sturmfels

TL;DR

This work develops a metric algebraic framework for the Grassmannian of lines by mapping skew-symmetric matrices to projection coordinates and measuring distance via the Frobenius norm. It introduces the Grassmann distance degree (GD degree) as the algebraic count of critical points in the distance optimization problem, and contrasts it with the Euclidean (ED) and maximum likelihood (ML) degrees, highlighting the role of the algebraic cut locus and extraneous critical points. The authors derive critical-equation machinery, analyze the base loci of the squaring map, and provide explicit GD and ED computations for lines in 3-space and for Schubert varieties, including a Chow threefold example that exhibits GDdeg = 10 versus EDdeg = 42. They also formulate conjectures for Schubert varieties and report extensive data on ED, GD, and SD degrees, illustrating that GD generally remains small even when ED can be large, thereby offering a practical algebraic approach to distance optimization on Gr(2,n).

Abstract

The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Plücker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.

Distance Optimization in the Grassmannian of Lines

TL;DR

This work develops a metric algebraic framework for the Grassmannian of lines by mapping skew-symmetric matrices to projection coordinates and measuring distance via the Frobenius norm. It introduces the Grassmann distance degree (GD degree) as the algebraic count of critical points in the distance optimization problem, and contrasts it with the Euclidean (ED) and maximum likelihood (ML) degrees, highlighting the role of the algebraic cut locus and extraneous critical points. The authors derive critical-equation machinery, analyze the base loci of the squaring map, and provide explicit GD and ED computations for lines in 3-space and for Schubert varieties, including a Chow threefold example that exhibits GDdeg = 10 versus EDdeg = 42. They also formulate conjectures for Schubert varieties and report extensive data on ED, GD, and SD degrees, illustrating that GD generally remains small even when ED can be large, thereby offering a practical algebraic approach to distance optimization on Gr(2,n).

Abstract

The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Plücker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.
Paper Structure (6 sections, 17 theorems, 53 equations, 1 figure)

This paper contains 6 sections, 17 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Squaring Skew-Symmetric Matrices
  3. Critical Equations
  4. The Grassmann Distance Degree
  5. Lines in 3-Space
  6. Schubert Varieties

Key Result

Theorem 2.1

The variety $\mathcal{V}_{X^2}$ has the expected dimension $\binom{n}{2}- 1$. It is the Zariski closure of the set of symmetric diagonalizable matrices whose nonzero eigenvalues have multiplicity $2$. The fiber of $\,(-)^2$ containing a generic rank $2r$ skew-symmetric $n \times n$ matrix $X$ has si

Figures (1)

  • Figure 1: The spectral region comprises all eigenvalue pairs $(\lambda,\mu)$ as $X$ ranges over $\mathcal{M}$. The marked points are the optimal solutions for the nine metrics on ${\rm Gr}(2,5)$ in the table above.

Theorems & Definitions (63)

  • Example 1.1: Schubert surface
  • Theorem 2.1
  • Lemma 2.2: gantmacher
  • proof : Proof of Theorem \ref{['thm:skew-sym']}
  • Corollary 2.3
  • proof
  • Example 2.4: $n=3$
  • Example 2.5: $n=4$
  • Example 2.6: $n = 5,6,7$
  • Conjecture 2.7
  • ...and 53 more