Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees

Recently there is a line of research work proposing to employ Spectral Clustering (SC) to segment (group){Throughout the paper, we use segmentation, clustering, and grouping, and their verb forms, interchangeably.} high-dimensional structural data such as those (approximately) lying on subspaces {We follow {liu2010robust} and use the term "subspace" to denote both linear subspaces and affine subspaces. There is a trivial conversion between linear subspaces and affine subspaces as mentioned therein.} or low-dimensional manifolds. By learning the affinity matrix in the form of sparse reconstruction, techniques proposed in this vein often considerably boost the performance in subspace settings where traditional SC can fail. Despite the success, there are fundamental problems that have been left unsolved: the spectrum property of the learned affinity matrix cannot be gauged in advance, and there is often one ugly symmetrization step that post-processes the affinity for SC input. Hence we advocate to enforce the symmetric positive semidefinite constraint explicitly during learning (Low-Rank Representation with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it can be solved in an exquisite scheme efficiently instead of general-purpose SDP solvers that usually scale up poorly. We provide rigorous mathematical derivations to show that, in its canonical form, LRR-PSD is equivalent to the recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting future research. As per the computational cost, our proposal is at most comparable to that of LRR, if not less. We validate our theoretic analysis and optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified and typos corrected. Experimental data slightly adde

Shape Interaction Matrix Revisited and Robustified: Efficient Subspace Clustering with Corrupted and Incomplete Data

The Shape Interaction Matrix (SIM) is one of the earliest approaches to performing subspace clustering (i.e., separating points drawn from a union of subspaces). In this paper, we revisit the SIM and reveal its connections to several recent subspace clustering methods. Our analysis lets us derive a simple, yet effective algorithm to robustify the SIM and make it applicable to realistic scenarios where the data is corrupted by noise. We justify our method by intuitive examples and the matrix perturbation theory. We then show how this approach can be extended to handle missing data, thus yielding an efficient and general subspace clustering algorithm. We demonstrate the benefits of our approach over state-of-the-art subspace clustering methods on several challenging motion segmentation and face clustering problems, where the data includes corrupted and missing measurements.Comment: This is an extended version of our iccv15 pape