International Journal of Computer Vision

, Volume 100, Issue 3, pp 217-240

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Hybrid Linear Modeling via Local Best-Fit Flats

  • Teng ZhangAffiliated withSchool of Mathematics, University of Minnesota
  • , Arthur SzlamAffiliated withCourant Institute of Mathematical Sciences, New York University
  • , Yi WangAffiliated withSchool of Mathematics, University of Minnesota
  • , Gilad LermanAffiliated withSchool of Mathematics, University of Minnesota Email author 


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’ β 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.


Hybrid linear modeling Subspace clustering Spectral clustering Local PCA High-dimensional data Motion segmentation Face clustering