  The bipartite graph structure has shown its promising ability in facilitating
the subspace clustering and spectral clustering algorithms for large-scale
datasets. To avoid the post-processing via k-means during the bipartite graph
partitioning, the constrained Laplacian rank (CLR) is often utilized for
constraining the number of connected components (i.e., clusters) in the
bipartite graph, which, however, neglects the distribution (or normalization)
of these connected components and may lead to imbalanced or even ill clusters.
Despite the significant success of normalized cut (Ncut) in general graphs, it
remains surprisingly an open problem how to enforce a one-step normalized cut
for bipartite graphs, especially with linear-time complexity. In this paper, we
first characterize a novel one-step bipartite graph cut (OBCut) criterion with
normalized constraints, and theoretically prove its equivalence to a trace
maximization problem. Then we extend this cut criterion to a scalable subspace
clustering approach, where adaptive anchor learning, bipartite graph learning,
and one-step normalized bipartite graph partitioning are simultaneously modeled
in a unified objective function, and an alternating optimization algorithm is
further designed to solve it in linear time. Experiments on a variety of
general and large-scale datasets demonstrate the effectiveness and scalability
of our approach.
