Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

Hannah Friedman, Serkan Hoşten

TL;DR

This work surveys Grassmannian and flag varieties through multiple algebraic realizations (Stiefel, Plücker, projection, isospectral) and studies the algebraic complexity of optimization problems posed on these spaces by counting complex critical points. It develops three equivalent formulations of the multi-eigenvector problem and related heterogeneous quadratic minimization, establishing exact critical-point counts via LO degrees and providing explicit descriptions in each coordinate system. The results extend to statistics problems—canonical correlation analysis and correspondence analysis—where optimality conditions reduce to singular-vector structures and yield precise combinatorial counts of critical points. Together, these contributions deepen understanding of optimization on flag varieties and supply concrete algebraic benchmarks for numerical methods and applications.

Abstract

Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure of the algebraic complexity is the amount of complex critical points of the corresponding optimization problem. After an exposition of different realizations of these manifolds as algebraic varieties we present a sample of optimization problems over them and we compute their algebraic complexity.

Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

TL;DR

This work surveys Grassmannian and flag varieties through multiple algebraic realizations (Stiefel, Plücker, projection, isospectral) and studies the algebraic complexity of optimization problems posed on these spaces by counting complex critical points. It develops three equivalent formulations of the multi-eigenvector problem and related heterogeneous quadratic minimization, establishing exact critical-point counts via LO degrees and providing explicit descriptions in each coordinate system. The results extend to statistics problems—canonical correlation analysis and correspondence analysis—where optimality conditions reduce to singular-vector structures and yield precise combinatorial counts of critical points. Together, these contributions deepen understanding of optimization on flag varieties and supply concrete algebraic benchmarks for numerical methods and applications.

Abstract

Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure of the algebraic complexity is the amount of complex critical points of the corresponding optimization problem. After an exposition of different realizations of these manifolds as algebraic varieties we present a sample of optimization problems over them and we compute their algebraic complexity.

Paper Structure

This paper contains 12 sections, 24 theorems, 66 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Many Lives of the Flag Variety
  3. The Multi-Eigenvector Problem
  4. Critical points in Stiefel coordinates
  5. Critical points in projection coordinates
  6. Critical points in isospectral coordinates
  7. Heterogeneous Quadratics Minimization Problem
  8. Diagonal case
  9. Projection coordinates
  10. Two Problems from Statistics
  11. Canonical correlation analysis
  12. Correspondence Analysis

Key Result

Theorem 1.1

Let $A$ be a generic real symmetric $n \times n$ matrix and let $Z$ be an $n \times k$ variable matrix. The algebraic set of complex critical points of the multi-eigenvector problem (multi-eigenvector) is where $u_1, \ldots, u_n$ is an orthonormal eigenbasis of $A$. This algebraic set is a disjoint union of $\binom{n}{k}$ varieties isomorphic to ${\rm O}(k)$; it has dimension $\dim({\rm O}(k)) =

Figures (1)

  • Figure 1: Diagram explaining how to move from one life of the flag variety to another. If $A \to B$ in the diagram, the edge label explains how to write the $B$ coordinates in terms of the $A$ coordinates. Two of the arrows are bidirectional, meaning that one direction comes from matrix multiplication and the other comes from a matrix factorization.

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: Proposition 9.1.1, fulton
  • Proposition 2.5: Theorem 5.2, DFRS
  • Proposition 2.6
  • Theorem 2.7
  • ...and 35 more