Least orthogonal absolute deviations

Nyquist, Hans

doi:10.1016/0167-9473(88)90076-x

Cited by 33 publications

(19 citation statements)

References 4 publications

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“…One cannot consider RSR without acknowledging work done on robust orthogonal regression and its subsequent extension to RSR [70,80,83,96,109]. In this problem, one fits a (D − 1)-dimensional subspace in R D , that is, an element of G(D, D − 1), using orthogonal distance as an error metric.…”

Section: J Parallel Workmentioning

confidence: 99%

An Overview of Robust Subspace Recovery

2018

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This paper will serve as an introduction to the body of work on robust subspace recovery. Robust subspace recovery involves finding an underlying low-dimensional subspace in a dataset that is possibly corrupted with outliers. While this problem is easy to state, it has been difficult to develop optimal algorithms due to its underlying nonconvexity. This work emphasizes advantages and disadvantages of proposed approaches and unsolved problems in the area.

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Section: J Parallel Workmentioning

confidence: 99%

An Overview of Robust Subspace Recovery

2018

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“…Even approximate minimization of this energy is nontrivial, since it has been shown to be NP hard for 1 ≤ p < 2 [12] (and assumed to be even harder for 0 < p < 1). This minimization was suggested with p = 1 and δ = 0 by Osborne and Watson [50], Späth and Watson [52], and Nyquist [49], who also proposed algorithmic solutions when d = D − 1. Later heuristic solutions were proposed for any d < D by Ding et al [17] and Zhang et al [67].…”

Section: The Underlying Minimization Problemmentioning

confidence: 99%

Fast, robust and non-convex subspace recovery

Maunu

2017

Information and Inference: A Journal of the IMA

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This work presents a fast and non-convex algorithm for robust subspace recovery. The data sets considered include inliers drawn around a low-dimensional subspace of a higher dimensional ambient space, and a possibly large portion of outliers that do not lie nearby this subspace. The proposed algorithm, which we refer to as Fast Median Subspace (FMS), is designed to robustly determine the underlying subspace of such data sets, while having lower computational complexity than existing methods. We prove convergence of the FMS iterates to a stationary point. Further, under a special model of data, FMS converges to a point which is near to the global minimum with overwhelming probability. Under this model, we show that the iteration complexity is globally bounded and locally r-linear. The latter theorem holds for any fixed fraction of outliers (less than 1) and any fixed positive distance between the limit point and the global minimum. Numerical experiments on synthetic and real data demonstrate its competitive speed and accuracy.

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“…This idea leads to the following optimization problem. In case d = D − 1, the problem (1.3) is sometimes called orthogonal 1 regression [SW87] or least orthogonal absolute deviations [Nyq88]. The extension to general d is apparently more recent [Wat01,DZHZ06].…”

Section: 2)mentioning

confidence: 99%

“…See [Har74a,Har74b,Dod87] for some historical discussion. It appears that orthogonal regression with LAD was first considered in the late 1980s [OW85,SW87,Nyq88]; the extension from orthogonal regression to PCA seems to be even more recent [Wat01,DZHZ06]. LAD has also been considered as a method for hybrid linear modeling in [ZSL09,LZ11].…”

Section: Previous Workmentioning

confidence: 99%

Robust Computation of Linear Models, or How to Find a Needle in a Haystack

McCoy¹,

Tropp²,

Zhang³

2012

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Abstract. Consider a dataset of vector-valued observations that consists of a modest number of noisy inliers, which are explained well by a low-dimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. 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 are contained in a low-dimensional subspace, there is a rigorous theory that describes when reaper can recover the subspace exactly.

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Least orthogonal absolute deviations

Cited by 33 publications

References 4 publications

An Overview of Robust Subspace Recovery

An Overview of Robust Subspace Recovery

Fast, robust and non-convex subspace recovery

Robust Computation of Linear Models, or How to Find a Needle in a Haystack

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