Analysis of Some Problems in Manifold Learning of LLE
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Aiming at the problem of nonlinear dimensionality reduction,presents local linear embedding(LLE)algorithm.The nonlinear structure in high dimensional data space is exploited with the local symmetries of linear reconstructions.Maps the data points in high dimensional space into corresponding data points in lower dimensional space under preserving distance between data points.Introduces a manifold learning algorithm of LLE,summarizes some problems of LLE and its research status,and discusses the prospect of LLE.Keywords:
Manifold (fluid mechanics)
Data point
Manifold alignment
Data space
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Locally Linear Embedding(LLE)algorithm is one of the nonlinear data dimensionality reduction approaches based on manifold learning. Considering the distribution of data points mostly present the heterogeneity, there will result in large amounts ofinformation loss when LLE selects neighboring points. This paper proposes a novel locally linear embedding algorithm based on tightness of data points, named tLLE, which can reduce dimensionality effectively for the datasets that present the non-uniform distribution. And, it has better effects of dimensionality reduction than LLE. The embedding and classification results on synthetic and real data show that tLLE is very effective.
Semidefinite embedding
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Manifold (fluid mechanics)
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Manifold (fluid mechanics)
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Manifold learning is a newer research direction of machine learning and congnitive science in recent years,its essence is to find out the low dimensional manifold hidden in high dimensional space though learning discrete samples,and get the hidden dimensional structure of the high dimensional data to realize non-linear dimension reduction.The paper introduced some manifold learning algorithms,summarized some problems of manifold learning and its research status,and discussed the prospect of manifold learning.
Manifold alignment
Manifold (fluid mechanics)
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Manifold (fluid mechanics)
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The locally linear embedding (LLE) algorithm is considered as a powerful method for the problem of nonlinear dimensionality reduction. In this paper, a new method called Self-Regulated LLE is proposed. It achieves to solve the problem of deciding appropriate neighborhood parameter for LLE by finding the local patch which is close to be a linear one. The experiment results show that LLE with self-regulation performs better in most cases than LLE based on different evaluation criteria and spends less time on several data sets.
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As a new kind of nonlinear dimensionality reduction method,manifold learning is capturing increasing interests of researchers.To understand manifold learning better,the principle is firstly introduced,and then its development history and different representations are summarized,finally several major method are introduced,whose basic thoughts,steps and advantages are pointed out respectively.By the experiments on Swiss-Roll,the selection of neighbors and noise effect are analyzed,the results shows:compared with traditional linear method,manifold learning can discover the intrinsic structure of the samples better.Finally the prospect of manifold learning was discussed for more developments.
Manifold (fluid mechanics)
Manifold alignment
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How to obtain the highly nonlinear low-dimensional manifolds in the high-dimensional observation space is the goal of manifold learning.Currently,most of the manifold learning algorithms are applied to the nonlinear dimensionality reduction and data visualization,such as Isomap,LLE,Laplacian Eigenmap etc.This paper analysises and compares this three manifold learning algorithms by experiments,which reveals the characteristics of manifold learning algorithms for dimensionality reduction and data analyses.
Isomap
Manifold alignment
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Finding meaningful low-dimensional embedded in a high-dimensional space is a classical problem. Isomap is a nonlinear dimensionality reduction method proposed and based on the theory of manifold. It not only can reveal the meaningful low-dimensional structure hidden in the high-dimensional observation data, but can recover the underlying parameter of data lying on a low-dimensional submanifold. Based on the hypothesis that there is an isometric mapping between the data space and the parameter space, Isomap works, but this hypothesis has not been proved. In this paper, the existence of isometric mapping between the manifold in the high-dimensional data space and the parameter space is proved. By distinguishing the intrinsic dimensionality of high-dimensional data space from the manifold dimensionality, and it is proved that the intrinsic dimensionality is the upper bound of the manifold dimensionality in the high-dimensional space in which there is a toroidal manifold. Finally an algorithm is proposed to find the underlying toroidal manifold and judge whether there exists one. The results of experiments on the multi-pose three-dimensional object show that the method is effective.
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