  The problem of clustering noisy and incompletely observed high-dimensional
data points into a union of low-dimensional subspaces and a set of outliers is
considered. The number of subspaces, their dimensions, and their orientations
are assumed unknown. We propose a simple low-complexity subspace clustering
algorithm, which applies spectral clustering to an adjacency matrix obtained by
thresholding the correlations between data points. In other words, the
adjacency matrix is constructed from the nearest neighbors of each data point
in spherical distance. A statistical performance analysis shows that the
algorithm exhibits robustness to additive noise and succeeds even when the
subspaces intersect. Specifically, our results reveal an explicit tradeoff
between the affinity of the subspaces and the tolerable noise level. We
furthermore prove that the algorithm succeeds even when the data points are
incompletely observed with the number of missing entries allowed to be (up to a
log-factor) linear in the ambient dimension. We also propose a simple scheme
that provably detects outliers, and we present numerical results on real and
synthetic data.
