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The Grassmannian of affine subspaces

Abstract

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore, it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics—linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components—may be naturally formulated as problems on the affine Grassmannian.

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Notes

  1. In certain areas of algebraic geometry and representation theory, notably Langland’s program, the term ‘affine Grassmannian’ widely refers to a functor associated with an algebraic group, which is completely unrelated to the sense in which it is used in this article.

  2. Definition 2 has appeared in [26, Definition 3.1]. We reproduce it here for the reader’s easy reference.

  3. A projection matrix satisfies \(P^2 = P\) and an orthogonal projection matrix is in addition symmetric, i.e., \(P^{\scriptscriptstyle {\mathsf {T}}}= P\). Despite its name, an orthogonal projection matrix P is not an orthogonal matrix unless P is an identity matrix.

  4. Definition 3 has appeared in [26, Definition 3.4]. We reproduce it here for the reader’s easy reference.

  5. A cell decomposition of a topological space X is a partition of X into a disjoint union of open subsets \(\{X_i\}_{i\in I}\) such that for each \(i\in I\) there is a continuous map \(f:B^{n_i} \rightarrow X\) from the unit closed ball \(B^{n_i}\) of dimension \(n_i\) to X satisfying (i) the restriction of f to the interior of \(B^{n_i}\) is a homeomorphism onto \(X_i\); and (ii) the image \(f(\partial B^{n_i})\) is contained in the union of finitely many \(X_j\)’s with \(\dim X_j < \dim X_i\).

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Acknowledgements

The authors thank the two referees for their very helpful comments and suggestions. In particular, Example 1 was suggested by one of them. The work in this article is supported by DARPA D15AP00109, NSF IIS 1546413, DMS 1209136, NSFC Grant no. 11801548, NSFC Grant no. 11688101 and National Key R&D Program of China Grant no. 2018YFA0306702. In addition, LHL’s work is supported by a DARPA Director’s Fellowship and the Eckhardt Faculty Fund; KY’s work is supported by the Hundred Talents Program of the Chinese Academy of Sciences and the Recruitment Program of Global Experts of China.

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Lim, LH., Wong, K.SW. & Ye, K. The Grassmannian of affine subspaces. Found Comput Math 21, 537–574 (2021). https://doi.org/10.1007/s10208-020-09459-8

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Keywords

  • Affine Grassmannian
  • Affine subspaces
  • Schubert calculus
  • homotopy and (co)homology
  • Probability densities
  • Distances and metrics
  • Multivariate data analysis

Mathematics Subject Classification

  • 14M15
  • 22F30
  • 46T12
  • 53C30
  • 57R22
  • 62H10