  Subspace segmentation or subspace learning is a challenging and complicated
task in machine learning. This paper builds a primary frame and solid
theoretical bases for the minimal subspace segmentation (MSS) of finite
samples. Existence and conditional uniqueness of MSS are discussed with
conditions generally satisfied in applications. Utilizing weak prior
information of MSS, the minimality inspection of segments is further simplified
to the prior detection of partitions. The MSS problem is then modeled as a
computable optimization problem via self-expressiveness of samples. A closed
form of representation matrices is first given for the self-expressiveness, and
the connection of diagonal blocks is then addressed. The MSS model uses a rank
restriction on the sum of segment ranks. Theoretically, it can retrieve the
minimal sample subspaces that could be heavily intersected. The optimization
problem is solved via a basic manifold conjugate gradient algorithm,
alternative optimization and hybrid optimization, taking into account of
solving both the primal MSS problem and its pseudo-dual problem. The MSS model
is further modified for handling noisy data, and solved by an ADMM algorithm.
The reported experiments show the strong ability of the MSS method on
retrieving minimal sample subspaces that are heavily intersected.
