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.
Among other subspace clustering methods, @cite_20 and @cite_27 are the only two papers that provide provable exact clustering guarantees for problems beyond independent subspaces (for which LRR provably gives dense graphs @cite_19 ). Their results however rely critically on the semi-random model assumption. For instance, @cite_27 uses the connectivity of a random k-nearest neighbor graph on a sphere to facilitate an argument for clustering consistency. In addition, these approaches do not easily generalize to SSC even under the semi-random model since the solution of SSC is considerably harder to characterize. In contrast, our results are much simpler and work generically without any probabilistic assumptions.
@cite_19 @cite_27 @cite_20
1781773254 2963840432 2615383372 2944309706
This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabelled input data points, which are assumed to lie in a union of low-dimensional subspaces. We show that a modified version of SSC is provably effective in correctly identifying the underlying subspaces, even with noisy data. This extends theoretical guarantee of this algorithm to the practical setting and provides justification to the success of SSC in a class of real applications. Subspace clustering methods based on l1, l2 or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, l1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, l2 and nuclear norm regularization provide efficient closed form solutions, but require very strong assumptions to guarantee a subspace-preserving affinity, e.g., independent subspaces and uncorrupted data. In this paper we study a subspace clustering method based on orthogonal matching pursuit. We show that the method is both computationally efficient and guaranteed to give a subspace-preserving affinity under broad conditions. Experiments on synthetic data verify our theoretical analysis, and applications in handwritten digit and face clustering show that our approach achieves the best trade off between accuracy and efficiency. Moreover, our approach is the first one to handle 100,000 data points. This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabeled input data points, which are assumed to be in a union of low-dimensional subspaces. We show that a modified version of SSC is provably effective in correctly identifying the underlying subspaces, even with noisy data. This extends theoretical guarantee of this algorithm to more practical settings and provides justification to the success of SSC in a class of real applications. 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”.