  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.
