  The problem of dimension reduction is of increasing importance in modern data
analysis. In this paper, we consider modeling the collection of points in a
high dimensional space as a union of low dimensional subspaces. In particular
we propose a highly scalable sampling based algorithm that clusters the entire
data via first spectral clustering of a small random sample followed by
classifying or labeling the remaining out of sample points. The key idea is
that this random subset borrows information across the entire data set and that
the problem of clustering points can be replaced with the more efficient and
robust problem of "clustering sub-clusters". We provide theoretical guarantees
for our procedure. The numerical results indicate we outperform other
state-of-the-art subspace clustering algorithms with respect to accuracy and
speed.
