Multidimensional Scaling in the Poincare disk

Cvetkovski, Andrej; Crovella, Mark

doi:10.18576/amis/100112

Cited by 14 publications

(10 citation statements)

References 16 publications

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“…For instance, low-distortion embedding of a general metric into a tree has been considered for various measures of distortion [1,6]. Low-distortion embedding of general metrics into hyperbolic spaces has also been considered by Walter et al [39,40] and Cvetkovski and Crovella [12]. In this paper, we use hyperbolic MDS for more truthful visualizations of tree variation.…”

Section: Low-distortion Embeddingsmentioning

confidence: 99%

“…Recently, Cvetkovski and Crovella [12] introduced a method MDS-PD (metric multidimensional scaling algorithm using the Poincaré disk model) which is based on a steepest decent method with hyperbolic line search. We adapted this software for our experiments with hyperbolic space, and we review the method here; more details can be found in [12]. Complex coordinates are used to present the points of the hyperbolic plane, making the Poincaré disk model a subset of the complex plane C: D = {z ∈ C||z| < 1}.…”

Section: Subtree Variance Correlationmentioning

confidence: 99%

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Quantification and Visualization of Variation in Anatomical Trees

Amenta

Datar

Dirksen³

et al. 2015

Association for Women in Mathematics Series

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This paper presents two approaches to quantifying and visualizing variation in datasets of trees. The first approach localizes subtrees in which significant population differences are found through hypothesis testing and sparse classifiers on subtree features. The second approach visualizes the global metric structure of datasets through low-distortion embedding into hyperbolic planes in the style of multidimensional scaling. A case study is made on a dataset of airway trees in relation to Chronic Obstructive Pulmonary Disease.

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Section: Low-distortion Embeddingsmentioning

confidence: 99%

Section: Subtree Variance Correlationmentioning

confidence: 99%

Quantification and Visualization of Variation in Anatomical Trees

Amenta

Datar

Dirksen³

et al. 2015

Association for Women in Mathematics Series

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“…In the linguistic field of distributional semantics, the learning of point configurations has been explored in [LW18, DSN + 18, TBG19]. The related problem of multidimensional scaling in hyperbolic space has also been treated in [LC78,WHPD14,SDSGR18,CC16]. The techniques required for many further applications of machine learning in hyperbolic space are developed in [GBH18].…”

Section: Related Workmentioning

confidence: 99%

Learning phylogenetic trees as hyperbolic point configurations

Wilson¹

2021

Preprint

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An alternative to independent pairwise distance estimation is proposed that uses hyperbolic geometry to jointly estimate pairwise distances subject to a weakening of the four point condition that characterises tree metrics. Specifically, taxa are represented as points in hyperbolic space such that the distance between a pair of points accounts for the site differences between the corresponding taxa. The proposed algorithm iteratively rearranges the points to increase an objective function that is shown empirically to increase the log-likelihood employed in tree search. Unlike the log-likelihood on tree space, the proposed objective function is differentiable, allowing for the use of gradient-based techniques in its optimisation. It is shown that the error term in the weakened four point condition is bounded by a linear function of the radius parameter of the hyperboloid model, which controls the curvature of the space. The error may thus be made as small as desired, within the bounds of computational precision.

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“…Indeed, the superiority of hyperbolic space for the embedding of tree-like graphs has been dramatically demonstrated in the recent works [NK17,DGRS18] 1 . The related problem of multi-dimensional scaling in hyperbolic space has also been approached in [LC78,WHPD14,DGRS18,CC16].…”

Section: Introductionmentioning

confidence: 99%

“…It suffers further from the drawback that the gradient of the distance function on the Poincaré ball is complicated to compute. Finally, [CC16] perform a steepest descent line search on the Poincaré disc using Möbius transformations, which is applicable only in the 2-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Gradient descent in hyperbolic space

Wilson,

Leimeister

2018

Preprint

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Gradient descent generalises naturally to Riemannian manifolds [Bon11], and to hyperbolic n-space, in particular. Namely, having calculated the gradient at the point on the manifold representing the model parameters, the updated point is obtained by travelling along the geodesic passing in the direction of the gradient. Some recent works employing optimisation in hyperbolic space have not attempted this procedure, however, employing instead various approximations to avoid a calculation that was considered to be too complicated. In this tutorial, we demonstrate that in the hyperboloid model of hyperbolic space, the necessary calculations to perform gradient descent are in fact straight-forward. The advantages of the approach are then both illustrated and quantified for the optimisation problem of computing the Fréchet mean (i.e. barycentre) of points in hyperbolic space.

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Multidimensional Scaling in the Poincare disk

Cited by 14 publications

References 16 publications

Quantification and Visualization of Variation in Anatomical Trees

Quantification and Visualization of Variation in Anatomical Trees

Learning phylogenetic trees as hyperbolic point configurations

Gradient descent in hyperbolic space

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