  We study the underlying structure of data (approximately) generated from a
union of independent subspaces. Traditional methods learn only one subspace,
failing to discover the multi-subspace structure, while state-of-the-art
methods analyze the multi-subspace structure using data themselves as the
dictionary, which cannot offer the explicit basis to span each subspace and are
sensitive to errors via an indirect representation. Additionally, they also
suffer from a high computational complexity, being quadratic or cubic to the
sample size. To tackle all these problems, we propose a method, called Matrix
Factorization with Column L0-norm constraint (MFC0), that can simultaneously
learn the basis for each subspace, generate a direct sparse representation for
each data sample, as well as removing errors in the data in an efficient way.
Furthermore, we develop a first-order alternating direction algorithm, whose
computational complexity is linear to the sample size, to stably and
effectively solve the nonconvex objective function and non- smooth l0-norm
constraint of MFC0. Experimental results on both synthetic and real-world
datasets demonstrate that besides the superiority over traditional and
state-of-the-art methods for subspace clustering, data reconstruction, error
correction, MFC0 also shows its uniqueness for multi-subspace basis learning
and direct sparse representation.
