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Schubert varieties and distances between subspaces of different dimensions. (English) Zbl 1365.14065
Authors’ 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.

14M15 Grassmannians, Schubert varieties, flag manifolds
15A18 Eigenvalues, singular values, and eigenvectors
14N20 Configurations and arrangements of linear subspaces
51K99 Distance geometry
Full Text: DOI arXiv
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