  Given full or partial information about a collection of points that lie close
to a union of several subspaces, subspace clustering refers to the process of
clustering the points according to their subspace and identifying the
subspaces. One popular approach, sparse subspace clustering (SSC), represents
each sample as a weighted combination of the other samples, with weights of
minimal $\ell_1$ norm, and then uses those learned weights to cluster the
samples. SSC is stable in settings where each sample is contaminated by a
relatively small amount of noise. However, when there is a significant amount
of additive noise, or a considerable number of entries are missing, theoretical
guarantees are scarce. In this paper, we study a robust variant of SSC and
establish clustering guarantees in the presence of corrupted or missing data.
We give explicit bounds on amount of noise and missing data that the algorithm
can tolerate, both in deterministic settings and in a random generative model.
Notably, our approach provides guarantees for higher tolerance to noise and
missing data than existing analyses for this method. By design, the results
hold even when we do not know the locations of the missing data; e.g., as in
presence-only data.
