  Sparse subspace clustering (SSC) is one of the current state-of-the-art
methods for partitioning data points into the union of subspaces, with strong
theoretical guarantees. However, it is not practical for large data sets as it
requires solving a LASSO problem for each data point, where the number of
variables in each LASSO problem is the number of data points. To improve the
scalability of SSC, we propose to select a few sets of anchor points using a
randomized hierarchical clustering method, and, for each set of anchor points,
solve the LASSO problems for each data point allowing only anchor points to
have a non-zero weight (this reduces drastically the number of variables). This
generates a multilayer graph where each layer corresponds to a different set of
anchor points. Using the Grassmann manifold of orthogonal matrices, the shared
connectivity among the layers is summarized within a single subspace. Finally,
we use $k$-means clustering within that subspace to cluster the data points,
similarly as done by spectral clustering in SSC. We show on both synthetic and
real-world data sets that the proposed method not only allows SSC to scale to
large-scale data sets, but that it is also much more robust as it performs
significantly better on noisy data and on data with close susbspaces and
outliers, while it is not prone to oversegmentation.
