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An efficient algorithm for Kernel two-dimensional principal component analysis

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Abstract

Recently, a new approach called two-dimensional principal component analysis (2DPCA) has been proposed for face representation and recognition. The essence of 2DPCA is that it computes the eigenvectors of the so-called image covariance matrix without matrix-to-vector conversion. Kernel principal component analysis (KPCA) is a non-linear generation of the popular principal component analysis via the Kernel trick. Similarly, the Kernelization of 2DPCA can be benefit to develop the non-linear structures in the input data. However, the standard K2DPCA always suffers from the computational problem for using the image matrix directly. In this paper, we propose an efficient algorithm to speed up the training procedure of K2DPCA. The results of experiments on face recognition show that the proposed algorithm can achieve much more computational efficiency and remarkably save the memory-consuming compared to the standard K2DPCA.

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Acknowledgments

This work was partly supported by the National Natural Science Foundations of China under grant 60503023, and partly supported by the Natural Science Foundations of Jiangsu province under the grant BK2005407, partly supported by the key laboratory of image processing and image communication of Jiangsu province under the grant ZK205013, and partly supported by Program for New Century Excellent Talents in University (NCET).

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Authors and Affiliations

  1. Research Center of Learning Science, Southeast University, Nanjing, 210096, China

    Ning Sun, Hai-xian Wang, Zhen-hai Ji & Li Zhao

  2. Department of Radio Engineering, Southeast University, Nanjing, 210096, China

    Ning Sun, Zhen-hai Ji, Cai-rong Zou & Li Zhao

Authors
  1. Ning Sun
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  2. Hai-xian Wang
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  3. Zhen-hai Ji
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  4. Cai-rong Zou
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  5. Li Zhao
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Corresponding author

Correspondence to Ning Sun.

Appendix A: centering samples in K2DPCA

Appendix A: centering samples in K2DPCA

From the analysis in the paper, the centralization of samples is equal to centralize the kernel matrix K. Let K C is the centered kernel matrix:

$$ K^{C} = BB^{T} $$
(A.1)

where

$$ B = {\left(\begin{aligned}\,& (\phi (x^{1}_{1}) - (1/N){\sum\nolimits_{i = 1}^N {\phi (x^{1}_{i})}})^{T} \\\,& \begin{array}{*{20}c} {{\begin{array}{*{20}c} & \\ \end{array}}}&& \\ \end{array} \vdots \\\,& (\phi (x^{1}_{N}) - (1/N){\sum\nolimits_{i = 1}^N {\phi (x^{1}_{i})}})^{T} \\\,& (\phi (x^{2}_{1}) - (1/N){\sum\nolimits_{i = 1}^N {\phi (x^{2}_{i})}})^{T} \\\,& \begin{array}{*{20}c} {{\begin{array}{*{20}c} & \\ \end{array}}}&& \\ \end{array} \vdots \\& (\phi (x^{s}_{N}) - (1/N){\sum\nolimits_{i = 1}^N {\phi (x^{s}_{i})}})^{T}. \\ \end{aligned} \right)} $$
(A.2)

Divide the K C into K C (j,l) j,l=1, ..., s , and the notation 1 N  is N ×  N matrix which all elements are equal to 1/N. Thus the centered kernel matrix is

$$ K^{C} (j,l) = K(j,l) - 1_{N} K(j,l) - K(j,l)1_{N} + 1_{N} K(j,l)1_{N}, \quad (j,l = 1, \ldots, s) $$
(A.3)

To the test samples, we centralize the test samples kernel matrix K(j) in (24). \({\hat{1}_{N} \,\hbox{is}\,s \times N}\) matrix and 1 N  is N ×  N matrix which all elements are equal to 1/N. So, the centered test samples kernel matrix K C (j) can be defined as following:

$$ \begin{aligned} K^{C} (j) &= {\left(\begin{aligned}\,& (\phi (t^{1}) - \frac{1}{N}{\sum\nolimits_{p = 1}^N {\phi (x^{1}_{p})}})^{T} \\\,& \begin{array}{*{20}c} &&& \\ \end{array} \vdots \\\,& (\phi (t^{s}) - \frac{1}{N}{\sum\nolimits_{p = 1}^N {\phi (x^{s}_{p})}})^{T} \\ \end{aligned} \right)}{\left({(\phi (x^{j}_{1}) - \frac{1}{N}{\sum\nolimits_{q = 1}^N {\phi (x^{j}_{q})), \ldots}},(\phi (x^{j}_{N}) - \frac{1}{N}{\sum\nolimits_{q = 1}^N {\phi (x^{j}_{q}))}})} \right)}\\ &= K(j) - K(j)1_{N} - \hat{1}_{N} \hat{K}(j) + \hat{1}_{N} \hat{K}(j)1_{N},\\ \end{aligned} $$
(A.4)

where \({\hat{K}(j)\, \hbox{is}\,N \times N}\) matrix and the element \({(\hat{K}(j))_{{pq}} }\) is

$$ (\hat{K}(j))_{{ab}} = \phi (x^{a}_{p})\phi (x^{j}_{b})^{T} $$
(A.5)

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Sun, N., Wang, Hx., Ji, Zh. et al. An efficient algorithm for Kernel two-dimensional principal component analysis. Neural Comput & Applic 17, 59–64 (2008). https://doi.org/10.1007/s00521-007-0111-0

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