Is an Affine Constraint Needed for Affine Subspace Clustering?

Chong You, Chun-Guang Li, Daniel Robinson, Rene Vidal

Abstract:

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 cluster-
ing methods can be applied directly. In motion segmenta-
tion, the subspaces are affine and an additional affine con-
straint on the coefficients is often enforced. However, since
affine subspaces can always be embedded into linear sub-
spaces of one extra dimension, it is unclear if the affine
constraint is really necessary. This paper shows, both the-
oretically 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 negligi-
ble effect on clustering performance. Specifically, our anal-
ysis 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 analy-
sis is the notion of affinely independent subspaces, which
not only provides geometrically interpretable correctness
conditions, but also clarifies the relationships between ex-
isting results for affine subspace clustering.