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Matrix Anal. Appl."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/15m1054201","type":"journal-article","created":{"date-parts":[[2016,9,1]],"date-time":"2016-09-01T14:04:42Z","timestamp":1472738682000},"page":"1176-1197","source":"Crossref","is-referenced-by-count":157,"title":["Schubert Varieties and Distances between Subspaces of Different Dimensions"],"prefix":"10.1137","volume":"37","author":[{"given":"Ke","family":"Ye","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lek-Heng","family":"Lim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,9,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"P.A. Absil, R. Mahony, and R. 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Van Loan,\n                      Matrix Computations\n                      , 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013."},{"key":"atypb21","first-page":"376","author":"Hamm J.","year":"2008","journal-title":"New York"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1111\/j.1420-9101.2008.01573.x"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"G. Haro, G. Randall, and G. Sapiro,\n                      Stratification learning: Detecting mixed density and dimensionality in high dimensional point clouds\n                      , in Advances in Neural Information Processing Systems (NIPS), Vol. 19, 2006, pp. 553-560.","DOI":"10.7551\/mitpress\/7503.003.0074"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2006.07.014"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"A. Ya. Helemskii,\n                      Lectures and Exercises on Functional Analysis\n                      , Transl. Math. 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