Consider a data set of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors and uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments that confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when reaper can approximate this subspace.
Bibliographical noteFunding Information:
Lerman and Zhang were supported in part by the IMA and by NSF Grants DMS-09-15064 and DMS-09-56072. McCoy and Tropp were supported by Office of Naval Research (ONR) Awards N00014-08-1-0883 and N00014-11-1002, Air Force Office of Scientific Research (AFOSR) Award FA9550-09-1-0643, Defense Advanced Research Projects Agency (DARPA) Award N66001-08-1-2065, and a Sloan Research Fellowship. The authors thank Eran Halperin, Yi Ma, Ben Recht, Amit Singer, and John Wright for helpful conversations. The anonymous referees provided many thoughtful and incisive remarks that helped us improve the manuscript immensely.
© 2014, SFoCM.
- Convex relaxation
- Iteratively reweighted least squares
- Robust linear models