Subspace clustering is the problem of clustering data points into a union of low-dimensional linear affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious "graph connectivity problem" (15). In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple post-processing procedure is capable of delivering consistent clustering under certain "general position" or "restricted eigenvalue" assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3.
An important problem in analyzing big data is subspace clustering , i.e., to represent a collection of points in a high-dimensional space via the union of low-dimensional subspaces. Sparse subspace clustering (SSC) and Low-rank representation (LRR) are the state-of-the-art methods for this task. These two methods are fundamentally similar in that both are based on convex optimization exploiting the intuition of “Self-Expressiveness”. The main difference is that the SSC minimizes the vector @math norm of the representation matrix to induce sparsity while LRR minimizes the nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-rank sparse subspace clustering (LRSSC), by the combining SSC and LRR, and develop theoretical guarantees of the success of the algorithm. The results reveal interesting insights into the strengths and the weaknesses of SSC and LRR, and demonstrate how the LRSSC can take advantage of both methods in preserving the “Self-Expressiveness Property” and “Graph Connectivity” at the same time. A byproduct of our analysis is that it also expands the theoretical guarantee of SSC to handle cases when the subspaces have arbitrarily small canonical angles but are “nearly independent”. Subspace clustering refers to the task of nding a multi-subspace representation that best ts a collection of points taken from a high-dimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) (25) to cluster noisy data, and develops some novel theory demonstrating its correctness. In particular, the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation, and on the number of samples per subspace. Synthetic as well as real data experiments complement our theoretical study, illustrating our approach and demonstrating its eectiveness. We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K-means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between subspaces. These conditions are weaker than those considered in the standard statistical literature. Experimental results on synthetic data generated from the standard unions of subspaces model demonstrate our theory. We also show that our algorithm performs competitively against state-of-the-art algorithms on real-world applications such as motion segmentation and face clustering, with much simpler implementation and lower computational cost.