Schubert varieties and distances between subspaces of different dimensions

2014/07/03 by Ke Ye, Lek‐Heng Lim, Ye, Ke +1 · 11 citations
Computer Science · Engineering · #14M15 #14N20 #15A18 #51K99 #Advanced Numerical Analysis Techniques #Advanced Vision and Imaging #Algebraic Geometry (math.AG) #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1407.0900

openalex publication_date 2014/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance --- if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.

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Related

• The Grassmann distance complexity
• Schubert Subspace Codes
• The Grassmannian of affine subspaces
• Asymmetric Geometry of Total Grassmannians
• Euclidean distance degree in manifold optimization