Minimum sum of distances estimator: robustness and stability

  • Authors:
  • Yoav Sharon;John Wright;Yi Ma

  • Affiliations:
  • Coordinated Science Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL;Coordinated Science Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL;Coordinated Science Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL

  • Venue:
  • ACC'09 Proceedings of the 2009 conference on American Control Conference
  • Year:
  • 2009

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Abstract

We consider the problem of estimating a state x from noisy and corrupted linear measurements y = Ax + z + e, where z is a dense vector of small-magnitude noise and e is a relatively sparse vector whose entries can be arbitrarily large. We study the behavior of the l1 estimator x = arg minx ||y - Ax||1, and analyze its breakdown point with respect to the number of corrupted measurements ||e||0. We show that the breakdown point is independent of the noise. We introduce a novel algorithm for computing the breakdown point for any given A, and provide a simple bound on the estimation error when the number of corrupted measurements is less than the breakdown point. As a motivational example we apply our algorithm to design a robust state estimator for an autonomous vehicle, and show how it can significantly improve performance over the Kalman filter.