  This paper studies the subspace clustering problem in which data points
collected from high-dimensional ambient space lie in a union of linear
subspaces. Subspace clustering becomes challenging when the dimension of
intersection between subspaces is large and most of the self-representation
based methods are sensitive to the intersection between the span of clusters.
In sharp contrast to the self-representation based methods, a recently proposed
clustering method termed Innovation Pursuit, computed a set of optimal
directions (directions of innovation) to build the adjacency matrix. This paper
focuses on the Innovation Pursuit Algorithm to shed light on its impressive
performance when the subspaces are heavily intersected. It is shown that in
contrast to most of the existing methods which require the subspaces to be
sufficiently incoherent with each other, Innovation Pursuit only requires the
innovative components of the subspaces to be sufficiently incoherent with each
other. These new sufficient conditions allow the clusters to be strongly close
to each other. Motivated by the presented theoretical analysis, a simple yet
effective projection based technique is proposed which we show with both
numerical and theoretical results that it can boost the performance of
Innovation Pursuit.
