On a Projected Weiszfeld Algorithm

Neumayer, Sebastian; Nimmer, Max; Steidl, Gabriele; Stephani, Henrike

doi:10.1007/978-3-319-58771-4_39

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…Local and asymptotic convergence rates of the Weiszfeld algorithm were given in [18] and a non-asymptotic sublinear convergence rate was proved in [3]. The very good performance of Weiszfeld's algorithm in comparison with the parallel proximal point algorithm was shown in [47] and a projected Weiszfeld algorithm was established in [38]. Keeling and Kunisch [20] suggested another stable algorithm for finding the geometric mean based on criticizing the behavior of the original Weiszfeld algorithm in anchor points and not taking its stabilized versions into account.…”

Section: Weiszfeld's Algorithm For Geometric Median Computationmentioning

confidence: 99%

On the Robust PCA and Weiszfeld’s Algorithm

Neumayer

Nimmer

Setzer³

et al. 2019

Appl Math Optim

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The principal component analysis (PCA) is a powerful standard tool for reducing the dimensionality of data. Unfortunately, it is sensitive to outliers so that various robust PCA variants were proposed in the literature. This paper addresses the robust PCA by successively determining the directions of lines having minimal Euclidean distances from the data points. The corresponding energy functional is not differentiable at a finite number of directions which we call anchor directions. We derive a Weiszfeld-like algorithm for minimizing the energy functional which has several advantages over existing algorithms. Special attention is paid to the careful handling of the anchor directions, where we take the relation between local minima and one-sided derivatives of Lipschitz continuous functions on submanifolds of R d into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka-Lojasiewicz property of the energy functional, we prove global convergence of the whole sequence of iterates generated by the algorithm to a critical point of the energy functional. Numerical examples demonstrate the very good performance of our algorithm.

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Section: Weiszfeld's Algorithm For Geometric Median Computationmentioning

confidence: 99%

On the Robust PCA and Weiszfeld’s Algorithm

Neumayer

Nimmer

Setzer³

et al. 2019

Appl Math Optim

Self Cite

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show abstract

“…Note the strong connection of this iterative scheme to the Weiszfeld algorithm [4,42,34] and majorize-minimize strategies [7]. By the following lemma, the gradient descent iteration (19) coincides with those of the conditional gradient algorithm (15) on the Grassmannian G d,K .…”

Section: Minimization Algorithmmentioning

confidence: 78%

“…Note the strong connection of this iterative scheme to the Weiszfeld algorithm [4,42,34] and majorize-minimize strategies [7].…”

Section: Hence P ⊥mentioning

confidence: 97%

On the rotational invariant L1-norm PCA

Neumayer

Nimmer

Setzer³

et al. 2020

Linear Algebra and its Applications

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Principal component analysis (PCA) is a powerful tool for dimensionality reduction. Unfortunately, it is sensitive to outliers, so that various robust PCA variants were proposed in the literature. One of the most frequently applied methods for high dimensional data reduction is the rotational invariant L 1 -norm PCA of Ding and coworkers. So far no convergence proof for this algorithm was available. The main topic of this paper is to fill this gap. We reinterpret this robust approach as a conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on Grassmannian manifolds. Based on the latter point of view, we prove global convergence of the whole series of iterates to a critical point using the Kurdyka-Lojasiewicz property of the objective function, where we have to pay special attention to so-called anchor points, where the function is not differentiable.

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Suggesting Waypoints to Autonomous Surface Vessels for Effective Emergency Response Using Machine Learning

Lekamge

Harischandra

2021

OCEANS 2021: San Diego – Porto

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On a Projected Weiszfeld Algorithm

Cited by 3 publications

References 17 publications

On the Robust PCA and Weiszfeld’s Algorithm

On the Robust PCA and Weiszfeld’s Algorithm

On the rotational invariant L1-norm PCA

Suggesting Waypoints to Autonomous Surface Vessels for Effective Emergency Response Using Machine Learning

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