3.8 Proceedings Paper

Is an Affine Constraint Needed for Affine Subspace Clustering?

Publisher

IEEE
DOI: 10.1109/ICCV.2019.01001

Keywords

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Funding

  1. National Science Foundation [1618637, 1704458]
  2. Northrop Grumman Corporation's Research in Applications for Learning Machines (REALM) Program
  3. National Natural Science Foundation of China [61876022]
  4. Open Project Fund from the Key Laboratory of Machine Perception (MOE), Peking University
  5. Direct For Computer & Info Scie & Enginr
  6. Division of Computing and Communication Foundations [1618637] Funding Source: National Science Foundation
  7. Div Of Information & Intelligent Systems
  8. Direct For Computer & Info Scie & Enginr [1704458] Funding Source: National Science Foundation

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Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really necessary. This paper shows, both theoretically and empirically, that when the dimension of the ambient space is high relative to the sum of the dimensions of the affine subspaces, the affine constraint has a negligible effect on clustering performance. Specifically, our analysis provides conditions that guarantee the correctness of affine subspace clustering methods both with and without the affine constraint, and shows that these conditions are satisfied for high-dimensional data. Underlying our analysis is the notion of affinely independent subspaces, which not only provides geometrically interpretable correctness conditions, but also clarifies the relationships between existing results for affine subspace clustering.

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