Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measurement, computation or privacy. In this paper, we study the theoretical properties of a popular subspace clustering algorithm named sparse subspace clustering (SSC) and establish formal success conditions of SSC on dimensionality-reduced data. Our analysis applies to the most general fully deterministic model where both underlying subspaces and data points within each subspace are deterministically positioned, and also a wide range of dimensionality reduction techniques (e.g., Gaussian random projection, uniform subsampling, sketching) that fall into a subspace embedding framework (Meng & Mahoney, 2013; , 2014). Finally, we apply our analysis to a differentially private SSC algorithm and established both privacy and utility guarantees of the proposed method.
This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizesthe standardlow-rankmatrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n×N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank ≤ r 1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.