Hybrid Linear Modeling via Local Best-fit Flats
Abstract
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' 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.
Keywords
Cite
@article{arxiv.1010.3460,
title = {Hybrid Linear Modeling via Local Best-fit Flats},
author = {Teng Zhang and Arthur Szlam and Yi Wang and Gilad Lerman},
journal= {arXiv preprint arXiv:1010.3460},
year = {2012}
}
Comments
This version adds some clarifications and numerical experiments as well as strengthens the previous theorem. For face experiments, we use here the Extended Yale Face Database B (cropped faces unlike previous version). This database points to a failure mode of our algorithms, but we suggest and successfully test a workaround