Published on Sat Feb 18 2012

Robust computation of linear models by convex relaxation

Gilad Lerman, Michael McCoy, Joel A. Tropp, Teng Zhang

The paper describes a convex optimization problem that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors. The paper provides an efficient algorithm for solving the problem.

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Abstract

Consider a dataset 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 it 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 which 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.