QUANTILE APPROXIMATION FOR ROBUST STATISTICAL ESTIMATION AND k-ENCLOSING PROBLEMS

Mount, David M.; Netanyahu, Nathan S.; Piatko, Christine D.; Silverman, Ruth; Wu, Angela Y.

doi:10.1142/s0218195900000334

Cited by 13 publications

(5 citation statements)

References 37 publications

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“…Mount et al (2007) present an algorithm based on branch and bound for p = 2 for computing approximate solutions to the LMS problem. Mount et al (2000) present a quantile approximation algorithm with approximation factor ε with complexity O(n log(n) + (1/ε) O(p) ). Chakraborty and Chaudhuri (2008) present probabilistic search algorithms for a class of problems in robust statistics.…”

Section: Related Workmentioning

confidence: 99%

Least quantile regression via modern optimization

Bertsimas¹,

Mazumder²

2014

Ann. Statist.

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We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_i,\mathbf{x}_i)$'s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n={}$10,000) LQS problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1223 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Related Workmentioning

confidence: 99%

Least quantile regression via modern optimization

Bertsimas¹,

Mazumder²

2014

Ann. Statist.

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“…The high complexity of the exact LMS computation motivated the development of many fast LMS approximation algorithms. The reader is referred to [11,13,14] and [15] for more detail.…”

Section: Lms Approximationsmentioning

confidence: 99%

GPU-Based Computation of 2D Least Median of Squares with Applications to Fast and Robust Line Detection

Shapira,

Hassner

2015

Preprint

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The 2D Least Median of Squares (LMS) is a popular tool in robust regression because of its high breakdown point: up to half of the input data can be contaminated with outliers without affecting the accuracy of the LMS estimator.The complexity of 2D LMS estimation has been shown to be Ω(n 2 ) where n is the total number of points. This high theoretical complexity along with the availability of graphics processing units (GPU) motivates the development of a fast, parallel, GPU-based algorithm for LMS computation. We present a CUDA based algorithm for LMS computation and show it to be much faster than the optimal state of the art single threaded CPU algorithm. We begin by describing the proposed method and analyzing its performance. We then demonstrate how it can be used to modify the well-known Hough Transform (HT) in order to efficiently detect image lines in noisy images. Our method is compared with standard HT-based line detection methods and shown to overcome their shortcomings in terms of both efficiency and accuracy.

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“…198–199] by updating the center and scatter estimates corresponding to the best ( p + 1)‐subset, using the h observations within its minimum volume ellipsoid. Several alternative algorithms to calculate the MVE have been proposed 17, 28–35…”

Section: Algorithmmentioning

confidence: 99%

Minimum volume ellipsoid

Aelst

Rousseeuw

2009

WIREs Computational Stats

161

104

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The minimum volume ellipsoid (MVE) estimator is based on the smallest volume ellipsoid that covers h of the n observations. It is an affine equivariant, highbreakdown robust estimator of multivariate location and scatter. The MVE can be computed by a resampling algorithm. Its low bias makes the MVE very useful for outlier detection in multivariate data, often through the use of MVE-based robust distances. We review the basic MVE definition as well as some useful extensions such as the one-step reweighted MVE. We discuss the main properties of the MVE including its breakdown value, affine equivariance, and efficiency. We discuss the basic resampling algorithm to calculate the MVE and illustrate its use on two examples. An overview of applications is given, as well as some related classes of robust estimators of multivariate location and scatter. 

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QUANTILE APPROXIMATION FOR ROBUST STATISTICAL ESTIMATION AND k-ENCLOSING PROBLEMS

Cited by 13 publications

References 37 publications

Least quantile regression via modern optimization

Least quantile regression via modern optimization

GPU-Based Computation of 2D Least Median of Squares with Applications to Fast and Robust Line Detection

Minimum volume ellipsoid

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