text: 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.
  @cite_15: This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which signicantly broadens the range of problems where it is provably eective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods.
  @cite_3: Sparse Subspace Clustering (SSC) is one of the recent approaches to subspace segmentation. In SSC a graph is constructed whose nodes are the data points and whose edges are inferred from the L1-sparse representation of each point by the others. It has been proved that if the points lie on a mixture of independent subspaces, the graphical structure of each subspace is disconnected from the others. However, the problem of connectivity within each subspace is still unanswered. This is important since the subspace segmentation in SSC is based on finding the connected components of the graph. Our analysis is built upon the connection between the sparse representation through L1-norm minimization and the geometry of convex poly-topes proposed by the compressed sensing community. After introduction of some assumptions to make the problem well-defined, it is proved that the connectivity within each subspace holds for 2- and 3-dimensional subspaces. The claim of connectivity for general d-dimensional case, even for generic configurations, is proved false by giving a counterexample in dimensions greater than 3.
summary: For the SSC, @cite_15 showed that if the samples are uniformly distributed in a union of subspaces @math and for the basis matrices @math of @math , @math with where @math is a give parameter, then SSC could give a block-diagonal solution partitioned as the ideal subspace segmentation with a probability approximately equal to one, depending on @math , @math , @math , and @math . We remark this claim does not imply a connected solution as we have explained in the early discussion or mentioned by @cite_3 .