Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. Existing algebraic subspace clustering methods are based on fitting the data with an algebraic variety and decomposing this variety into its constituent subspaces. Such methods are well suited to the case of a known number of subspaces of known and equal dimensions, where a single polynomial vanishing in the variety is sufficient to identify the subspaces. While subspaces of unknown and arbitrary dimensions can be handled using multiple vanishing polynomials, current approaches are not robust to corrupted data due to the difficulty of estimating the number of polynomials. As a consequence, the current practice is to use a single polynomial to fit the data with a union of hyperplanes containing the union of subspaces, an approach that works well only when the dimensions of the subspaces are high enough. In this paper, we propose a new algebraic subspace clustering algorithm, which can identify the subspace ...