Robust computation of linear models by convex relaxation

TL;DR

This paper introduces REAPER, a convex optimization method for robustly fitting low-dimensional linear models to data with outliers, supported by an efficient algorithm and theoretical guarantees.

Contribution

It presents a novel convex relaxation approach, REAPER, for robust subspace recovery, along with an efficient solver and theoretical analysis of its performance.

Findings

REAPER reliably identifies linear subspaces in noisy, outlier-contaminated data.

Numerical experiments demonstrate REAPER's effectiveness on synthetic and real datasets.

Theoretical results specify conditions under which REAPER accurately approximates the true subspace.

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.