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Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment [article]

Zhenyue Zhang, Hongyuan Zha
2002 arXiv   pre-print
In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction.  ...  Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood  ...  Principal Manifolds via Local Tangent Space Alignment  ... 
arXiv:cs/0212008v1 fatcat:24rnvy2klnfrhgxefdmgodmwpi

Principal manifolds and nonlinear dimensionality reduction via tangent space alignment

Zhen-yue Zhang, Hong-yuan Zha
2004 Journal of Shanghai University (English Edition)  
In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction.  ...  Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood  ...  Principal Manifolds via Local Tangent Space Alignment  ... 
doi:10.1007/s11741-004-0051-1 fatcat:q5rhi7ikh5hnfd2zv3k3zhbyee

Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

Zhenyue Zhang, Hongyuan Zha
2004 SIAM Journal on Scientific Computing  
In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction.  ...  Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood  ...  Principal Manifolds via Local Tangent Space Alignment  ... 
doi:10.1137/s1064827502419154 fatcat:gbyyh7ctmnbarnpbjggbon2ylu

Local coordinate weight reconstruction for rolling bearing fault diagnosis

Rong Jiang, Zhonghua Huang, Chenxi Wu, Xin Wu, Zhe Liu
2020 Journal of Vibroengineering  
The experimental results show that the intraclass aggregation and interclass differences of global low-dimensional coordinates extracted via LCWR are better than those of local tangent space alignment  ...  (LTSA), locally linear embedding (LLE) and principal component analysis (PCA).  ...  Xiangtan guiding science and technology plan project (ZDX-CG2019004).  ... 
doi:10.21595/jve.2020.21460 fatcat:sfqap4rxlvb3de4v5bhbdqcqgy

Linear local tangent space alignment and application to face recognition

Tianhao Zhang, Jie Yang, Deli Zhao, Xinliang Ge
2007 Neurocomputing  
In this paper, linear local tangent space alignment (LLTSA), as a novel linear dimensionality reduction algorithm, is proposed.  ...  It uses the tangent space in the neighborhood of a data point to represent the local geometry, and then aligns those local tangent spaces in the low-dimensional space which is linearly mapped from the  ...  Acknowledgments The authors would like to thank the anonymous reviewers and editors for their comments and suggestions, which helped to improve the quality of this paper greatly.  ... 
doi:10.1016/j.neucom.2006.11.007 fatcat:h7grcuz6nfds5hgawui7zchc2u

Incremental Manifold Learning Via Tangent Space Alignment [chapter]

Xiaoming Liu, Jianwei Yin, Zhilin Feng, Jinxiang Dong
2006 Lecture Notes in Computer Science  
In this paper, we proposed an incremental version (ILTSA) of LTSA (Local Tangent Space Alignment), which is one of the key manifold learning algorithms.  ...  They have been used to extract the intrinsic characteristic of different type of high-dimensional data by performing nonlinear dimensionality reduction.  ...  spaces, and these methods have been regarded as effective approaches for nonlinear dimension reduction.  ... 
doi:10.1007/11829898_10 fatcat:nzhlssfuqrat5na723zqfg2d6m

A Local Tangent Space Alignment Based Transductive Classification Algorithm [chapter]

Jianwei Yin, Xiaoming Liu, Zhilin Feng, Jinxiang Dong
2006 Lecture Notes in Computer Science  
LTSA (local tangent space alignment) is a recently proposed method for manifold learning, which can efficiently learn nonlinear embedding low-dimensional coordinates of high-dimensional data, and can also  ...  traditional LDA, then the global low-dimensional embedding manifold is obtained by local affine transforms, finally TCM-KNN method is used for classification on the low-dimensional manifold.  ...  LTSA Algorithm LTSA is a nonlinear dimension reduction algorithm operated on tangent space. Data are assumed to lie on noised nonlinear low-dimensional manifold in the algorithm.  ... 
doi:10.1007/11829898_9 fatcat:n6ki4q3aprgbffijydrmqempdq

Semi-supervised nonlinear dimensionality reduction

Xin Yang, Haoying Fu, Hongyuan Zha, Jesse Barlow
2006 Proceedings of the 23rd international conference on Machine learning - ICML '06  
It is shown that basic nonlinear dimensionality reduction algorithms, such as Locally Linear Embedding (LLE), Isometric feature mapping (ISOMAP), and Local Tangent Space Alignment (LTSA), can be modified  ...  The problem of nonlinear dimensionality reduction is considered. We focus on problems where prior information is available, namely, semi-supervised dimensionality reduction.  ...  Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM Journal on Scientific Computing, 26(1), 313-338.  ... 
doi:10.1145/1143844.1143978 dblp:conf/icml/YangFZB06 fatcat:vdlriqbzjrbh3li7ryewjgr2aq

Laplacian PCA and Its Applications

Deli Zhao, Zhouchen Lin, Xiaoou Tang
2007 2007 IEEE 11th International Conference on Computer Vision  
Manifold unfolding (non-linear dimensionality reduction) can be performed by the alignment of tangential maps which are linear transformations of tangent coordinates approximated by LPCA.  ...  The LPCA algorithm is based on the global alignment of locally Gaussian or linear subspaces via an alignment technique borrowed from manifold learning.  ...  Acknowledgement The authors would like to thank Yi Ma and John Wright for discussion, and Wei Liu for his comments.  ... 
doi:10.1109/iccv.2007.4409096 dblp:conf/iccv/ZhaoLT07a fatcat:pvcvstxktbfo7gyoy3unzeyvme

Nonlinear Dimensionality Reduction via the ENH-LTSA Method for Hyperspectral Image Classification

Weiwei Sun, Avner Halevy, John J. Benedetto, Wojciech Czaja, Weiyue Li, Chun Liu, Beiqi Shi, Rongrong Wang
2014 IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing  
The problems of neglecting spatial features in hyperspectral imagery (HSI) and the high complexity of Local Tangent Space Alignment (LTSA) still exist in the nonlinear dimensionality reduction with LTSA  ...  First, random projection is introduced to preliminarily reduce the dimension of HSI data. It aims to improve the speed of neighbor searching and the local tangent space construction.  ...  To make a holistic analysis, ENH-LTSA is compared with other state-of-the-art dimensionality reduction methods, LTSA, LLTSA (Linear Local Tangent Space Alignment) [50] , PCA, and LE.  ... 
doi:10.1109/jstars.2013.2238890 fatcat:s4dcdvkpdfbsdlpbnt2754gr5a

Representing Edge Models via Local Principal Component Analysis [chapter]

Patrick S. Huggins, Steven W. Zucker
2002 Lecture Notes in Computer Science  
Representing Edge Models via Local Principal Component Analysis 385 edge model E with parameter space Θ, we seek the 'best' explanation of I, e.g., findingθ ∈ Θ that maximizes p(E(θ)|I).  ...  The parameter estimation problem can be viewed geometrically [27] [3]: the edge model E is a low-dimensional manifold embedded in a high-dimensional space, where the dimensionality is given by the number  ...  The principal components approximate the local tangent space to the manifold, enabling better reconstruction than in (b).  ... 
doi:10.1007/3-540-47969-4_26 fatcat:ibh6gfydmrhgrdrp2jhbukk7mq

Riemannian Manifold Clustering and Dimensionality Reduction for Vision-Based Analysis [chapter]

Alvina Goh
2011 Advances in Computer Vision and Pattern Recognition  
To address these problems, algorithms for performing simultaneous nonlinear dimensionality reduction and clustering of data sampled from multiple submanifolds of a Riemannian manifold have been recently  ...  Over the past few years, various techniques have been developed for learning a low-dimensional representation of a nonlinear manifold embedded in a high-dimensional space.  ...  Review of Local Nonlinear Dimensionality Reduction Methods in Euclidean Spaces In this section, we review three local nonlinear dimensionality reduction algorithms for data lying in a single manifold.  ... 
doi:10.1007/978-0-85729-057-1_2 fatcat:txv4ivisjja7jb4r65eswz32ne

Kernel Principal Geodesic Analysis [chapter]

Suyash P. Awate, Yen-Yun Yu, Ross T. Whitaker
2014 Lecture Notes in Computer Science  
Kernel principal component analysis (kPCA) has been proposed as a dimensionality-reduction technique that achieves nonlinear, low-dimensional representations of data via the mapping to kernel feature space  ...  It then applies these tools to propose novel methods for (i) dimensionality reduction and (ii) clustering using mixture-model fitting.  ...  Nonlinear Dimensionality Reduction We employ kPCA and the proposed kPGA for nonlinear dimensionality reduction on simulated and real-world databases.  ... 
doi:10.1007/978-3-662-44848-9_6 fatcat:usqo3y4amnayhj5fldlzaubuke

Towards view-invariant expression analysis using analytic shape manifolds

Sima Taheri, Pavan Turaga, Rama Chellappa
2011 Face and Gesture 2011  
We use landmark configurations to represent facial deformations and exploit the fact that the affine shape-space can be studied using the Grassmann manifold.  ...  We extend some of the available approaches for expression analysis to the Grassmann manifold and experimentally show promising results, paving the way for a more general theory of view-invariant expression  ...  APPENDIX Here we present the solutions to some problems related to traversing the Grassmann manifold which will be of use in expression analysis.  ... 
doi:10.1109/fg.2011.5771415 dblp:conf/fgr/TaheriTC11 fatcat:l3mm6k3745bebo53rro3ud2v3u

A geometric viewpoint of manifold learning

Binbin Lin, Xiaofei He, Jieping Ye
2015 Applied Informatics  
The discussion is focused on the problem of dimensionality reduction and semi-supervised learning.  ...  The manifold assumption, which states that the data is sampled from a submanifold embedded in much higher dimensional Euclidean space, has been widely adopted by many researchers.  ...  These methods try to find coordinate representation for curved manifolds. LTSA tries to construct a global coordinate via local tangent space alignment.  ... 
doi:10.1186/s40535-015-0006-6 fatcat:zzbifclg5nf7tkqxxjap3bkxsq
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