Robust computation of linear models by convex relaxation
This addresses the challenge of robust linear modeling for data analysis in noisy environments, representing an incremental improvement with a rigorous theoretical guarantee.
The paper tackles the problem of robustly fitting a low-dimensional subspace to data with noisy inliers and outliers by introducing REAPER, a convex optimization method, which reliably identifies linear structure in synthetic and natural data.
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.