---
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.'
address:
- 'Department of Mathematics, University of Chicago, Chicago, IL 60637.'
- 'Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL 60637.'
author:
- Ke Ye
- 'Lek-Heng Lim'
title: Schubert varieties and distances between subspaces of different dimensions
---

Introduction {#sec:motivation}
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Biological data (e.g. gene expression levels, metabolomic profile), image data (e.g.  