Optimal Algorithms for &lt;formula formulatype="inline"&gt; &lt;tex Notation="TeX"&gt;$L_{1}$&lt;/tex&gt;&lt;/formula&gt;-subspace Signal Processing

Markopoulos, Panos P.; Karystinos, George N.; Pados, Dimitris A.

doi:10.1109/tsp.2014.2338077

Cited by 142 publications

(7 citation statements)

References 38 publications

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“…However, because the L2-norm enlarges the influence of outliers, the traditional functional principal components analysis method is sensitive to outliers. On the other hand, in regard to multivariate data, relevant research of principal component analysis methods [30][31][32][33][34][35][36][37] has shown that the principal component analysis method of L1-norm for multivariate data has a better robustness than that of the L2-norm. In [30], Kwak (2008) proposed an L1-PCA optimization model based on L1-norm maximization for multivariable data, i.e., W L1 = argmax…”

Section: Introductionmentioning

confidence: 99%

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A Method of L1-Norm Principal Component Analysis for Functional Data

Liu

et al. 2020

Symmetry

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Recently, with the popularization of intelligent terminals, research on intelligent big data has been paid more attention. Among these data, a kind of intelligent big data with functional characteristics, which is called functional data, has attracted attention. Functional data principal component analysis (FPCA), as an unsupervised machine learning method, plays a vital role in the analysis of functional data. FPCA is the primary step for functional data exploration, and the reliability of FPCA plays an important role in subsequent analysis. However, classical L2-norm functional data principal component analysis (L2-norm FPCA) is sensitive to outliers. Inspired by the multivariate data L1-norm principal component analysis methods, we propose an L1-norm functional data principal component analysis method (L1-norm FPCA). Because the proposed method utilizes L1-norm, the L1-norm FPCs are less sensitive to the outliers than L2-norm FPCs which are the characteristic functions of symmetric covariance operator. A corresponding algorithm for solving the L1-norm maximized optimization model is extended to functional data based on the idea of the multivariate data L1-norm principal component analysis method. Numerical experiments show that L1-norm FPCA proposed in this paper has a better robustness than L2-norm FPCA, and the reconstruction ability of the L1-norm principal component analysis to the original uncontaminated functional data is as good as that of the L2-norm principal component analysis.

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Section: Introductionmentioning

confidence: 99%

“…where d = rank(X); this solution has a low performance degradation, and is close to L2-PCA, but the cost is that it is not as robust as that in [30]. The work in [32] offered an algorithm for exact calculation…”

Section: Introductionmentioning

confidence: 99%

A Method of L1-Norm Principal Component Analysis for Functional Data

Liu

et al. 2020

Symmetry

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“…In recent years, low-rank matrix recovery [24][25][26][27][28][29][30][31][32][33][34][35][36] has attracted much attention in the areas of face recognition, image recovery, de-noising, and data reconstruction. The main purpose of low-rank matrix reconstruction is to recover the low-rank matrix from a large, but sparse error data set.…”

Section: Introductionmentioning

confidence: 99%

“…In different applications, this is also called sparse and low-rank matrix decomposition, robust principle component analysis (RPCA), and rank-sparsity incoherence. The traditional low-rank matrix approximation methods only consider the low-rank approximation of a single matrix, such as RPCA [24], principal component analysis based on 1 l -norm (PCAL1) [25,26], and the recently developed low-rank matrix theory [27,28]. These methods can be considered as generalized cases of the sparse representation (SR) [29,30] method.…”

Section: Introductionmentioning

confidence: 99%

Block Sparse Low-rank Matrix Decomposition based Visual Defect Inspection of Rail Track Surfaces

2019

KSII TIIS

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Low-rank matrix decomposition has shown its capability in many applications such as image in-painting, de-noising, background reconstruction and defect detection etc. In this paper, we consider the texture background of rail track images and the sparse foreground of the defects to construct a low-rank matrix decomposition model with block sparsity for defect inspection of rail tracks, which jointly minimizes the nuclear norm and the 2-1 norm. Similar to ADM, an alternative method is proposed in this study to solve the optimization problem. After image decomposition, the defect areas in the resulting low-rank image will form dark stripes that horizontally cross the entire image, indicating the preciselocations of the defects. Finally, a two-stage defect extraction method is proposed to locate the defect areas. The experimental results of the two datasets show that our algorithm achieved better performance compared with other methods.

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“…First, apply the principal component analysis on the high dimensional vectors to perform the dimension reductions for the machine learning and the data analysis applications [8], [20]. This includes the pattern recognition applications [15] of the subspace mapping [1] and the subspace learning [14]. For example, it can be applied to the optimal design of the agile supply chain network under an uncertainty [12] and the reduction of the total number of the dimensions of a logistic regression model with the continuous covariates and the avoidance of the multi-collinearity [4].…”

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confidence: 99%

Principal component analysis with drop rank covariance matrix

Guo¹,

Ling²

2021

JIMO

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This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

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Optimal Algorithms for <formula formulatype="inline"> <tex Notation="TeX">$L_{1}$</tex></formula>-subspace Signal Processing

Cited by 142 publications

References 38 publications

A Method of L1-Norm Principal Component Analysis for Functional Data

A Method of L1-Norm Principal Component Analysis for Functional Data

Block Sparse Low-rank Matrix Decomposition based Visual Defect Inspection of Rail Track Surfaces

Principal component analysis with drop rank covariance matrix

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