Outlier detection and least trimmed squares approximation using semi-definite programming

Nguyen, Tri-Dung; Welsch, Roy E.

doi:10.1016/j.csda.2009.09.037

Cited by 43 publications

(31 citation statements)

References 19 publications

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“…Perhaps the closest in spirit to our paper is the work of Nguyen and Welsch [12], who tailored a maxmin formulation for outlier removal in least squares regression. Their objective function takes the following form:…”

Section: Related Workmentioning

confidence: 99%

“…Invoking the minimax theorem [15, Corollary 37.3.2] allows us to swap min s,w and max π in (5) without changing the optimal object value. Setting = 0 and replacing f i (w) in (5b) with the convex and continuous squared cost as used in (12), we obtain an equivalent reformulation of (12). Squaring f i (w) in (5b) produces a set of quadratic constraints, which can be easily converted to Second Order Cone constraints.…”

Section: Related Workmentioning

confidence: 99%

“…One possible remedy for the computational issue of the SDP-based method is to use our minmax formulation (5) to rewrite (12) as an equivalent but less expensive SecondOrder Cone Program (SOCP). The equivalence of (5) and (12) is derived from the fact that the min s,w and max π operations in (5) are interchangeable if f i (w) is continuous and convex.…”

Section: Related Workmentioning

confidence: 99%

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An Adversarial Optimization Approach to Efficient Outlier Removal

Eriksson

Chin

et al. 2013

J Math Imaging Vis

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This paper proposes a novel adversarial optimization approach to efficient outlier removal in computer vision. We characterize the outlier removal problem as a game that involves two players of conflicting interests, namely, optimizer and outlier. Such an adversarial view not only brings new insights into various existing methods, but also gives rise to a general optimization framework that provably unifies them. Under the proposed framework, we develop a new outlier removal approach that is able to offer a much needed control over the trade-off between reliability and speed, which is otherwise not available in previous methods. The proposed approach is driven by a mixed-integer minmax (convex-concave) optimization process. Although a minmax problem is generally not amenable to efficient optimization, we show that for some commonly used vision objective functions, an equivalent Linear Program reformulation exists. We demonstrate our method on two representative multiview geometry problems. Experiments on real image data illustrate superior practical performance of our method over recent techniques.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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An Adversarial Optimization Approach to Efficient Outlier Removal

Eriksson

Chin

et al. 2013

J Math Imaging Vis

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“…Giloni and Padberg (2002) were the first to show this property, and used it to devise a local minimization procedure. Nguyen and Welsch (2010) revisited this formulation and derived an SDP formulation of the corresponding maximization problem. Unfortunately, the degeneracy of the feasible domain makes it difficult to apply concave minimization algorithms to problem (LTS).…”

Section: Formulation As a Best Subset Problemmentioning

confidence: 99%

“…In fact, Nguyen and Welsch (2010) showed that Problem (8) can be cast as an SDP problem, therefore it can be solved using standard widely-available software. If d + 1 ≤ q ≤ n and we force the variables to be binary, the solution to Problem (8) is the subset of q observation with the largest sum of squared residuals.…”

Section: Obtaining Good Approximate Boundsmentioning

confidence: 99%

SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression

Flores

2015

European Journal of Operational Research

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This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, tipically d ≤ 5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.

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