Hybrid Linear Modeling via Local Best-fit Flats

In this paper we present a simple and fast geometric method for modeling data by a union of affine sets. The method begins by forming a collection of local best fit affine subspaces. The correct sizes of the local neighborhoods are determined automatically by the Jones' $\beta_2$ numbers; we prove under certain geometric conditions that good local neighborhoods exist and are found by our method. The collection 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.