Sparse Subspace Clustering: Algorithm, Theory, and Applications

Abstract

We propose and study an algorithm, called Sparse Subspace Clustering, to cluster high-dimensional data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points that come from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data. As solving the sparse optimization program is NP-hard, we consider its convex relaxation and show that, under appropriate conditions on the arrangement of the subspaces and the distribution of the data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm does not require initialization, can be solved efficiently, and can handle data points near the intersections of subspaces. In addition, our algorithm can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by modifying the optimization program to incorporate the model of the data. We verify the effectiveness of the proposed algorithm through experiments on synthetic data as well as two real-world problems of motion segmentation and face clustering.