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.
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 .
@cite_15 @cite_3
2139054653 2054199929
Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of components. Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an @math matrix @math and a rank parameter @math . In our first algorithm, @math is chosen, and we let @math , where @math is the Moore-Penrose generalized inverse of @math . In our second algorithm @math , @math , @math are chosen, and we let @math . ( @math and @math are matrices that consist of actual columns and rows, respectively, of @math , and @math is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least @math , @math , where @math is the best rank- @math approximation provided by truncating the SVD of @math , and where @math is the Frobenius norm of the matrix @math . The number of columns of @math and rows of @math is a low-degree polynomial in @math , @math , and @math . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, The subspace approximation problem Subspace( @math , @math ) asks for a @math -dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the @math norm instead. Most of the previous work on subspace approximation has focused on small or constant @math and @math , using coresets and sampling techniques from computational geometry. In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, , with the approximation guarantee roughly @math , where @math is the @math -th moment of a standard normal random variable N(0,1). We show that the convex relaxation we use has an integrality gap (or "rank gap") of @math , for any constant @math . Finally, we show that assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than @math , for any constant @math .