  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; Avron et al., 2014). Finally, we apply our analysis to a
differentially private SSC algorithm and established both privacy and utility
guarantees of the proposed method.
