  We present a simple and fast geometric method for modeling data by a union of
affine subspaces. The method begins by forming a collection of local best-fit
affine subspaces, i.e., subspaces approximating the data in local
neighborhoods. The correct sizes of the local neighborhoods are determined
automatically by the Jones' $\beta_2$ numbers (we prove under certain geometric
conditions that our method finds the optimal local neighborhoods). The
collection of subspaces is further processed by a greedy selection procedure or
a spectral method to generate the final model. We discuss applications to
tracking-based motion segmentation and clustering of faces under different
illuminating conditions. We give extensive experimental evidence demonstrating
the state of the art accuracy and speed of the suggested algorithms on these
problems and also on synthetic hybrid linear data as well as the MNIST
handwritten digits data; and we demonstrate how to use our algorithms for fast
determination of the number of affine subspaces.
