Preservation of local linearity by neighborhood subspace scaling for solving the pre-image problem

An important issue involved in kernel methods is the pre-image problem. However, it is an ill-posed problem, as the solution is usually nonexistent or not unique. In contrast to direct methods aimed at minimizing the distance in feature space, indirect methods aimed at constructing approximate equiv...

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Bibliographic Details
Published inFrontiers of information technology & electronic engineering Vol. 15; no. 4; pp. 254 - 264
Main Authors Yang, Sheng-kai, Meng, Jian-yi, Shen, Hai-bin
Format Journal Article
LanguageEnglish
Published Heidelberg Zhejiang University Press 01.04.2014
Springer Nature B.V
Subjects
Algorithms
Communications Engineering
Computer Hardware
Computer Science
Computer Systems Organization and Communication Networks
Electrical Engineering
Electronics and Microelectronics
Ill posed problems
Instrumentation
Iterative methods
Kernel functions
Linearity
Mapping
Networks
Principal components analysis
Transformations (mathematics)
Local linearity preserving
Kernel PCA
Nonlinear denoising
TP391
Kernel method
Pre-image problem
Online AccessGet full text
ISSN1869-1951
2095-9184
1869-196X
2095-9230
DOI10.1631/jzus.C1300248

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Summary:An important issue involved in kernel methods is the pre-image problem. However, it is an ill-posed problem, as the solution is usually nonexistent or not unique. In contrast to direct methods aimed at minimizing the distance in feature space, indirect methods aimed at constructing approximate equivalent models have shown outstanding performance. In this paper, an indirect method for solving the pre-image problem is proposed. In the proposed algorithm, an inverse mapping process is constructed based on a novel framework that preserves local linearity. In this framework, a local nonlinear transformation is implicitly conducted by neighborhood subspace scaling transformation to preserve the local linearity between feature space and input space. By extending the inverse mapping process to test samples, we can obtain pre-images in input space. The proposed method is non-iterative, and can be used for any kernel functions. Experimental results based on image denoising using kernel principal component analysis （PCA） show that the proposed method outperforms the state-of-the-art methods for solving the pre-image problem.
Bibliography:Kernel method, Pre-image problem, Nonlinear denoising, Kernel PCA, Local linearity preserving
An important issue involved in kernel methods is the pre-image problem. However, it is an ill-posed problem, as the solution is usually nonexistent or not unique. In contrast to direct methods aimed at minimizing the distance in feature space, indirect methods aimed at constructing approximate equivalent models have shown outstanding performance. In this paper, an indirect method for solving the pre-image problem is proposed. In the proposed algorithm, an inverse mapping process is constructed based on a novel framework that preserves local linearity. In this framework, a local nonlinear transformation is implicitly conducted by neighborhood subspace scaling transformation to preserve the local linearity between feature space and input space. By extending the inverse mapping process to test samples, we can obtain pre-images in input space. The proposed method is non-iterative, and can be used for any kernel functions. Experimental results based on image denoising using kernel principal component analysis （PCA） show that the proposed method outperforms the state-of-the-art methods for solving the pre-image problem.
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ISSN:1869-1951
2095-9184
1869-196X
2095-9230
DOI:10.1631/jzus.C1300248