  In this paper, we propose a low-rank representation with symmetric constraint
(LRRSC) method for robust subspace clustering. Given a collection of data
points approximately drawn from multiple subspaces, the proposed technique can
simultaneously recover the dimension and members of each subspace. LRRSC
extends the original low-rank representation algorithm by integrating a
symmetric constraint into the low-rankness property of high-dimensional data
representation. The symmetric low-rank representation, which preserves the
subspace structures of high-dimensional data, guarantees weight consistency for
each pair of data points so that highly correlated data points of subspaces are
represented together. Moreover, it can be efficiently calculated by solving a
convex optimization problem. We provide a rigorous proof for minimizing the
nuclear-norm regularized least square problem with a symmetric constraint. The
affinity matrix for spectral clustering can be obtained by further exploiting
the angular information of the principal directions of the symmetric low-rank
representation. This is a critical step towards evaluating the memberships
between data points. Experimental results on benchmark databases demonstrate
the effectiveness and robustness of LRRSC compared with several
state-of-the-art subspace clustering algorithms.
