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On the Least Trimmed Squares Estimator

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Abstract

The linear least trimmed squares (LTS) estimator is a statistical technique for fitting a linear model to a set of points. Given a set of n points in ℝd and given an integer trimming parameter h≤n, LTS involves computing the (d−1)-dimensional hyperplane that minimizes the sum of the smallest h squared residuals. LTS is a robust estimator with a 50 %-breakdown point, which means that the estimator is insensitive to corruption due to outliers, provided that the outliers constitute less than 50 % of the set. LTS is closely related to the well known LMS estimator, in which the objective is to minimize the median squared residual, and LTA, in which the objective is to minimize the sum of the smallest 50 % absolute residuals. LTS has the advantage of being statistically more efficient than LMS. Unfortunately, the computational complexity of LTS is less understood than LMS. In this paper we present new algorithms, both exact and approximate, for computing the LTS estimator. We also present hardness results for exact and approximate LTS. A number of our results apply to the LTA estimator as well.

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Notes

  1. We use the asymptotic forms \(\widetilde{O}\) and \(\widetilde {\varOmega}\) as a shorthand for O and Ω, respectively, where factors of the form logO(1) n have been ignored.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Maryland, College Park, MD, USA

    David M. Mount

  2. Department of Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Nathan S. Netanyahu

  3. Center for Automation Research, University of Maryland, College Park, MD, USA

    Nathan S. Netanyahu & Ruth Silverman

  4. The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA

    Christine D. Piatko

  5. Department of Computer Science, American University, Washington, DC, USA

    Angela Y. Wu

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  1. David M. Mount
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  2. Nathan S. Netanyahu
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  3. Christine D. Piatko
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  4. Ruth Silverman
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  5. Angela Y. Wu
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Correspondence to David M. Mount.

Additional information

Dedicated to the memory of our dear friend and longtime colleague, Ruth Silverman.

D.M. Mount is partially supported by NSF grant CCF-1117259 and ONR grant N00014-08-1-1015.

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Mount, D.M., Netanyahu, N.S., Piatko, C.D. et al. On the Least Trimmed Squares Estimator. Algorithmica 69, 148–183 (2014). https://doi.org/10.1007/s00453-012-9721-8

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  • DOI: https://doi.org/10.1007/s00453-012-9721-8

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