Tensor Principal Component Analysis

Tensor Principal Component Analysis

David Zhang (Hong Kong Polytechnic University, Hong Kong), Fengxi Song (New Star Research Institute Of Applied Technology, China), Yong Xu (Harbin Institute of Technology, China) and Zhizhen Liang (Shanghai Jiao Tong University, China)
DOI: 10.4018/978-1-60566-200-8.ch008
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Abstract

Tensor principal component analysis (PCA) is an effective method for data reconstruction and recognition. In this chapter, some variants of classical PCA are introduced and the properties of tensor PCA are analyzed. Section 8.1 gives the background and development of tensor PCA. Section 8.2 introduces tensor PCA. Section 8.3 discusses some potential applications of tensor PCA in biometrics. Finally, we summarize this chapter in Section 8.4.
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Introduction

Principal component analysis (Turk & Pentland, 1991; Penev & Sirovich, 2000), also known as Karhunen-Loève (K-L) transform, is a classical statistical technique that has been widely used in various fields, such as face recognition, character recognition, and knowledge representation. The aim of PCA is to reduce the dimensionality of the data so that the extracted features are representative as possible. In general, the key idea of PCA is to project data to an orthogonal subspace, which can transform correlated variables into a smaller number of uncorrelated variables. The first principal component can capture variance of the data along some direction as possible, and consequent components capture as much of the remaining variability as possible. Up to now, there are a number of theoretical analyses and discussion for PCA in the literature and PCA is one of the most popular methods for data representation.

In recent years, some researchers (Yang, Zhang, Frangi, & Yang, 2004; Ye, 2004) noted that classical PCA often runs up against computational limits due to the high time and space complexity for dealing with large image matrices, especially for images and videos. In applying PCA, data must be converted to a vector form. This results in the difficulty in eigen-decomposition in a high dimensional vector space. To overcome this limitation, a novel idea is developed. This novel idea lies in dealing with image matrices or video data directly rather than converting them into vectors prior to dimensionality reduction. Based on this, Yang, Zhang, Frangi and Yang (2004) proposed a two-dimensional PCA for image representation, whose idea is that 2D image matrices are used to directly construct the image covariance matrix. This improves the computational efficiency. Moreover, the projection of sample on each principal orthogonal vector is a vector. A drawback of 2DPCA is that it needs more coefficients than PCA for image representation and costs more time to calculate distance in classification phase. In order to address this problem, Ye (2004) proposed a new algorithm called generalized low rank approximations of matrices (GLRAM) to reduce the computational cost. Then some researchers proposed a non-iterative algorithm for GLRAM (Liang & Shi, 2005; Liang, Zhang, & Shi, 2007). Moreover, they reveal the optimal property of GLRAM and show that the reconstruction errors of GLRAM are not smaller than those of PCA when considering the same dimension. Likewise, their method is proved to have much less computational time than the traditional singular value decomposition (SVD) technique. In addition, researchers also developed a number of variants of 2DPCA (Xu et al., 2005; Nhat & Lee, 2005; Xu, Jin, Jiang, & Guo, 2006; Hou, Gao, Pan, & Zhang, 2006; Vasilescu & Terzopoulos, 2002; Wang & Ahuja, 2004; Zuo Wang, & Zhang, 2005a, 2005b). In fact, the methods mentioned above belong to the framework proposed by Lathauwer and his partners (Lathauwer, 1997; Lathauwer, Moor, & Vandewalle, 2000a). In 2000, a multilinear generalization of the singular value decomposition was further proposed (Lathauwer, Moor, & Vandewalle, 2000b). Moreover, they also analyzed some properties of the matrix and the higher-order tensor decompositions. Yu and Bennamoun (2006) also proposed nD-PCA algorithm which exploits higher-order singular value decomposition. All the methods contribute to the development of the tensor PCA.

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Table of Contents
Acknowledgment
Chapter 1
Overview  (pages 1-23)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
A biometric system can be regarded as a pattern recognition system. In this chapter, we discuss two advanced pattern recognition technologies for... Sample PDF
Overview
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Chapter 2
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
This chapter is a brief introduction to biometric discriminant analysis technologies — Section I of the book. Section 2.1 describes two kinds of... Sample PDF
Discriminant Analysis for Biometric Recognition
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Chapter 3
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
As mentioned in Chapter II, there are two kinds of LDA approaches: classification- oriented LDA and feature extraction-oriented LDA. In most... Sample PDF
Discriminant Criteria for Pattern Classification
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Chapter 4
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we first give a brief introduction to Fisher linear discriminant, Foley- Sammon discriminant, orthogonal component discriminant... Sample PDF
Orthogonal Discriminant Analysis Methods
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Chapter 5
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we mainly present three kinds of weighted LDA methods. In Sections 5.1, 5.2 and 5.3, we respectively present parameterized direct... Sample PDF
Parameterized Discriminant Analysis Methods
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Chapter 6
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we introduce two novel facial feature extraction methods. The first is multiple maximum scatter difference (MMSD) which is an... Sample PDF
Two Novel Facial Feature Extraction Methods
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Chapter 7
Tensor Space  (pages 135-149)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we first give the background materials for developing tensor discrimination technologies in Section 7.1. Section 7.2 introduces... Sample PDF
Tensor Space
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Chapter 8
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
Tensor principal component analysis (PCA) is an effective method for data reconstruction and recognition. In this chapter, some variants of... Sample PDF
Tensor Principal Component Analysis
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Chapter 9
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
Linear discriminant analysis is a very effective and important method for feature extraction. In general, image matrices are often transformed into... Sample PDF
Tensor Linear Discriminant Analysis
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Chapter 10
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we describe two tensor-based subspace analysis approaches (tensor ICA and tensor NMF) that can be used in many fields like face... Sample PDF
Tensor Independent Component Analysis and Tensor Non-Negative Factorization
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Chapter 11
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we describe tensor-based classifiers, tensor canonical correlation analysis and tensor partial least squares, which can be used in... Sample PDF
Other Tensor Analysis and Further Direction
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Chapter 12
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In the past decades while biometrics attracts increasing attention of researchers, people also have found that the biometric system using a single... Sample PDF
From Single Biometrics to Multi-Biometrics
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Chapter 13
Feature Level Fusion  (pages 273-304)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
This chapter introduces the basis of feature level fusion and presents two feature level fusion examples. As the beginning, Section 13.1 provides an... Sample PDF
Feature Level Fusion
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Chapter 14
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With this chapter we aims at describing several basic aspects of matching score level fusion. Section 14.1 provides a description of basic... Sample PDF
Matching Score Level Fusion
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Chapter 15
Decision Level Fusion  (pages 328-348)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With this chapter, we first present a variety of decision level fusion rules and classifier selection approaches, and then show a case study of face... Sample PDF
Decision Level Fusion
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Chapter 16
Book Summary  (pages 349-358)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With the title “Advanced Pattern Recognition Technologies with Applications to Biometrics” this book mainly focuses on two kinds of advanced... Sample PDF
Book Summary
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About the Authors