A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics. Gajamannage, K., Paffenroth, R., & Bollt, E., M. 7, 2017. Paper Website abstract bibtex Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques in the presence of noise is not guaranteed. In fact, the embedding generated using such non-smooth, noisy measurements may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements have been corrupted by noise. Our method generates a network structure for given high-dimensional data using a neighborhood search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for noisy and sparse datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.
@article{
title = {A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics},
type = {article},
year = {2017},
identifiers = {[object Object]},
websites = {http://arxiv.org/abs/1707.06757},
month = {7},
day = {21},
id = {9b313392-21f7-349d-bc0f-2e0d47780523},
created = {2017-09-07T16:45:17.636Z},
accessed = {2017-09-07},
file_attached = {true},
profile_id = {5db6d3e7-562f-3ec2-a249-16ecf1e747e4},
group_id = {49665d18-5720-3154-b3f7-40652b55b7b9},
last_modified = {2017-09-07T16:45:34.085Z},
read = {false},
starred = {false},
authored = {false},
confirmed = {false},
hidden = {false},
folder_uuids = {9d95a352-241f-4364-baa0-2d8ae0f3f55d},
private_publication = {false},
abstract = {Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques in the presence of noise is not guaranteed. In fact, the embedding generated using such non-smooth, noisy measurements may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements have been corrupted by noise. Our method generates a network structure for given high-dimensional data using a neighborhood search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for noisy and sparse datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.},
bibtype = {article},
author = {Gajamannage, Kelum and Paffenroth, Randy and Bollt, Erik M.}
}
Downloads: 0
{"_id":"sRGidqazjmphqBn3p","bibbaseid":"gajamannage-paffenroth-bollt-anonlineardimensionalityreductionframeworkusingsmoothgeodesics-2017","authorIDs":[],"author_short":["Gajamannage, K.","Paffenroth, R.","Bollt, E., M."],"bibdata":{"title":"A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics","type":"article","year":"2017","identifiers":"[object Object]","websites":"http://arxiv.org/abs/1707.06757","month":"7","day":"21","id":"9b313392-21f7-349d-bc0f-2e0d47780523","created":"2017-09-07T16:45:17.636Z","accessed":"2017-09-07","file_attached":"true","profile_id":"5db6d3e7-562f-3ec2-a249-16ecf1e747e4","group_id":"49665d18-5720-3154-b3f7-40652b55b7b9","last_modified":"2017-09-07T16:45:34.085Z","read":false,"starred":false,"authored":false,"confirmed":false,"hidden":false,"folder_uuids":"9d95a352-241f-4364-baa0-2d8ae0f3f55d","private_publication":false,"abstract":"Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques in the presence of noise is not guaranteed. In fact, the embedding generated using such non-smooth, noisy measurements may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements have been corrupted by noise. Our method generates a network structure for given high-dimensional data using a neighborhood search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for noisy and sparse datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.","bibtype":"article","author":"Gajamannage, Kelum and Paffenroth, Randy and Bollt, Erik M.","bibtex":"@article{\n title = {A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics},\n type = {article},\n year = {2017},\n identifiers = {[object Object]},\n websites = {http://arxiv.org/abs/1707.06757},\n month = {7},\n day = {21},\n id = {9b313392-21f7-349d-bc0f-2e0d47780523},\n created = {2017-09-07T16:45:17.636Z},\n accessed = {2017-09-07},\n file_attached = {true},\n profile_id = {5db6d3e7-562f-3ec2-a249-16ecf1e747e4},\n group_id = {49665d18-5720-3154-b3f7-40652b55b7b9},\n last_modified = {2017-09-07T16:45:34.085Z},\n read = {false},\n starred = {false},\n authored = {false},\n confirmed = {false},\n hidden = {false},\n folder_uuids = {9d95a352-241f-4364-baa0-2d8ae0f3f55d},\n private_publication = {false},\n abstract = {Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques in the presence of noise is not guaranteed. In fact, the embedding generated using such non-smooth, noisy measurements may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements have been corrupted by noise. Our method generates a network structure for given high-dimensional data using a neighborhood search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for noisy and sparse datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.},\n bibtype = {article},\n author = {Gajamannage, Kelum and Paffenroth, Randy and Bollt, Erik M.}\n}","author_short":["Gajamannage, K.","Paffenroth, R.","Bollt, E., M."],"urls":{"Paper":"https://bibbase.org/service/mendeley/4b66b327-35ad-3956-a9a2-307331dd9988/file/356dad5e-e562-804d-5f73-851d0eb1a36a/full_text.pdf.pdf","Website":"http://arxiv.org/abs/1707.06757"},"bibbaseid":"gajamannage-paffenroth-bollt-anonlineardimensionalityreductionframeworkusingsmoothgeodesics-2017","role":"author","downloads":0},"bibtype":"article","creationDate":"2020-04-29T21:11:31.714Z","downloads":0,"keywords":[],"search_terms":["nonlinear","dimensionality","reduction","framework","using","smooth","geodesics","gajamannage","paffenroth","bollt"],"title":"A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics","year":2017}