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Spherical and hyperbolic embeddings of data

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Spherical and hyperbolic embeddings of data. / Wilson, Richard Charles; Hancock, Edwin R; Pekalska, Elzbieta; Duin, Robert P. W.

In: IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 36, No. 11, 11.04.2014, p. 2255-2268.

Research output: Contribution to journalArticlepeer-review

Harvard

Wilson, RC, Hancock, ER, Pekalska, E & Duin, RPW 2014, 'Spherical and hyperbolic embeddings of data', IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 36, no. 11, pp. 2255-2268. https://doi.org/10.1109/TPAMI.2014.2316836

APA

Wilson, R. C., Hancock, E. R., Pekalska, E., & Duin, R. P. W. (2014). Spherical and hyperbolic embeddings of data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(11), 2255-2268. https://doi.org/10.1109/TPAMI.2014.2316836

Vancouver

Wilson RC, Hancock ER, Pekalska E, Duin RPW. Spherical and hyperbolic embeddings of data. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2014 Apr 11;36(11):2255-2268. https://doi.org/10.1109/TPAMI.2014.2316836

Author

Wilson, Richard Charles ; Hancock, Edwin R ; Pekalska, Elzbieta ; Duin, Robert P. W. / Spherical and hyperbolic embeddings of data. In: IEEE Transactions on Pattern Analysis and Machine Intelligence. 2014 ; Vol. 36, No. 11. pp. 2255-2268.

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@article{a9fcc078c33c40869df77c0de9b81d6f,
title = "Spherical and hyperbolic embeddings of data",
abstract = "Many computer vision and pattern recognition problems may be posed as the analysis of a set of {\bf dissimilarities} between objects. For many types of data, these dissimilarities are not Euclidean (i.e. they do not represent the distances between points in a Euclidean space), and therefore cannot be isometrically embedded in a Euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-Euclidean data onto surfaces of constant curvature. Weaim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical)or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating theobjects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the Euclidean tangent space. This process is efficient enough to allow us to embed datasets of several thousand objects.We apply our method to a variety of data including time warping functions,shape similarities, graph similarity and gesture similarity data. In each case theembedding maintains the local structure of the data while placing the points in a metric space.",
keywords = "embedding, non-Euclidean, spherical, hyperbolic",
author = "Wilson, {Richard Charles} and Hancock, {Edwin R} and Elzbieta Pekalska and Duin, {Robert P. W.}",
year = "2014",
month = apr,
day = "11",
doi = "10.1109/TPAMI.2014.2316836",
language = "English",
volume = "36",
pages = "2255--2268",
journal = "IEEE Transactions on Pattern Analysis and Machine Intelligence",
issn = "0162-8828",
publisher = "IEEE Computer Society",
number = "11",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Spherical and hyperbolic embeddings of data

AU - Wilson, Richard Charles

AU - Hancock, Edwin R

AU - Pekalska, Elzbieta

AU - Duin, Robert P. W.

PY - 2014/4/11

Y1 - 2014/4/11

N2 - Many computer vision and pattern recognition problems may be posed as the analysis of a set of {\bf dissimilarities} between objects. For many types of data, these dissimilarities are not Euclidean (i.e. they do not represent the distances between points in a Euclidean space), and therefore cannot be isometrically embedded in a Euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-Euclidean data onto surfaces of constant curvature. Weaim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical)or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating theobjects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the Euclidean tangent space. This process is efficient enough to allow us to embed datasets of several thousand objects.We apply our method to a variety of data including time warping functions,shape similarities, graph similarity and gesture similarity data. In each case theembedding maintains the local structure of the data while placing the points in a metric space.

AB - Many computer vision and pattern recognition problems may be posed as the analysis of a set of {\bf dissimilarities} between objects. For many types of data, these dissimilarities are not Euclidean (i.e. they do not represent the distances between points in a Euclidean space), and therefore cannot be isometrically embedded in a Euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-Euclidean data onto surfaces of constant curvature. Weaim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical)or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating theobjects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the Euclidean tangent space. This process is efficient enough to allow us to embed datasets of several thousand objects.We apply our method to a variety of data including time warping functions,shape similarities, graph similarity and gesture similarity data. In each case theembedding maintains the local structure of the data while placing the points in a metric space.

KW - embedding

KW - non-Euclidean

KW - spherical

KW - hyperbolic

U2 - 10.1109/TPAMI.2014.2316836

DO - 10.1109/TPAMI.2014.2316836

M3 - Article

VL - 36

SP - 2255

EP - 2268

JO - IEEE Transactions on Pattern Analysis and Machine Intelligence

JF - IEEE Transactions on Pattern Analysis and Machine Intelligence

SN - 0162-8828

IS - 11

ER -