  Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.
