  In this work, we address the following matrix recovery problem: suppose we
are given a set of data points containing two parts, one part consists of
samples drawn from a union of multiple subspaces and the other part consists of
outliers. We do not know which data points are outliers, or how many outliers
there are. The rank and number of the subspaces are unknown either. Can we
detect the outliers and segment the samples into their right subspaces,
efficiently and exactly? We utilize a so-called {\em Low-Rank Representation}
(LRR) method to solve this problem, and prove that under mild technical
conditions, any solution to LRR exactly recovers the row space of the samples
and detect the outliers as well. Since the subspace membership is provably
determined by the row space, this further implies that LRR can perform exact
subspace segmentation and outlier detection, in an efficient way.
