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Algebraic Clustering of Affine Subspaces.

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Title: Algebraic Clustering of Affine Subspaces.
Authors: Tsakiris, Manolis C.1, Vidal, Rene1
Source: IEEE Transactions on Pattern Analysis & Machine Intelligence. Feb2018, Vol. 40 Issue 2, p482-489. 8p.
Subjects: Machine learning, Subspaces (Mathematics), Image analysis, Affine geometry, Polynomials
Abstract: Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces. [ABSTRACT FROM PUBLISHER]
Copyright of IEEE Transactions on Pattern Analysis & Machine Intelligence is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22IEEE+Transactions+on+Pattern+Analysis+%26+Machine+Intelligence%22">IEEE Transactions on Pattern Analysis & Machine Intelligence</searchLink>. Feb2018, Vol. 40 Issue 2, p482-489. 8p.
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  Data: Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces. [ABSTRACT FROM PUBLISHER]
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  Data: <i>Copyright of IEEE Transactions on Pattern Analysis & Machine Intelligence is the property of IEEE and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1109/TPAMI.2017.2678477
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      – Code: eng
        Text: English
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        PageCount: 8
        StartPage: 482
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      – SubjectFull: Machine learning
        Type: general
      – SubjectFull: Subspaces (Mathematics)
        Type: general
      – SubjectFull: Image analysis
        Type: general
      – SubjectFull: Affine geometry
        Type: general
      – SubjectFull: Polynomials
        Type: general
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      – TitleFull: Algebraic Clustering of Affine Subspaces.
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              Text: Feb2018
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