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Principal Component Analysis Based on L1-Norm Maximization Nojun Kwak IEEE Transactions on Pattern Analysis and Machine Intelligence, 2008

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Outline Introduction Background Knowledge Problem Description Algorithms Experiments Conclusion 2

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Introduction In data analysis problems, why do we need dimensionality reduction? Principal Component Analysis(PCA) PCA based on the L2-Norm is prone to the presence of outliers. 3

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Introduction Some algorithms for this problem: – L1-PCA Weighted median method Convex programming method Maximum likelihood estimation method – R1-PCA 4

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Background Knowledge L1-Norm, L2-Norm Principal Component Analysis(PCA) 5

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Lp-Norm Consider an n-dimensional vector: Define the p-Norm: L1-Norm is L2-Norm is 6

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Lp-Norm For example, x = [1, 2, 3] Special case : 7 namesymbolvalueapproximation L1-Norm|x| 1 66.000 L2-Norm|x| 2 3.742 L3-Norm|x| 3 3.302 L4-Norm|x| 4 3.146 L ∞ -Norm|x| ∞ 33.000

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Principal Component Analysis Principal component analysis (PCA) is a technique to seek projections that best preserve the data in a least-squares sense. The projections constitute a low-dimensional linear subspace. 8

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Principal Component Analysis The projection vectors, …, are the eigenvectors of the scatter matrix having the largest eigenvalues. 9 Scatter matrix:

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Principal Component Analysis The rotational invariance property: a fundamental property of Euclidean space with L2-Norm. So, PCA has rotational invariance property. 10

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Problem Description Traditional PCA: the presence of outliers. The effect of the outliers with a large norm is exaggerated by the use of the L2-Norm. Other method? 11

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Problem Description If we use L1-Norm instead of L2-Norm: where is the dataset. 12 is the projection matrix. is the coefficient matrix.

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Problem Description However, it’s very hard to achieve the exact solution. To resolve it, Ding et al. propose the R1-Norm and an approximate solution. 13 We call it R1-PCA.

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Problem Description The solution of R1-PCA depends on the dimension of subspace being found. The optimal solution when is not necessarily a subspace of when. The proposed method: PCA-L1 14

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We consider that: The maximization is done on the feature space. Algorithms 15 ensure to orthonormality of the projection matrix.

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Algorithms However, it’s difficult to find a global solution for. The optimal ith projection varies with different as in R1-PCA. How to solve it? 16

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Algorithms We simplify it into a series of problems using a greedy search method. Then, if we set, it become that: 17 Although the successive greedy solutions may differ from the optimal solution, it’s expected to provide a good approximation.

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Algorithms The optimization is still difficult because it contains absolute value operation, which is nonlinear. 18

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Algorithms 19

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Algorithms However, does the PCA-L1 procedure finds a local maximum point ? We should prove it. 20

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Theorem Theorem: With the PCA-L1 procedure vector converges to, which is a local maximum points of. The proof includes two parts: – is a non-decreasing function of. – The objective function has a local maximum value at. 21

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Proof is a non-decreasing function of. is the set of optimal polarity corresponding to. For all, 22

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Proof This holds because 23 are parallel. The inner product of two vectors.

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Proof So, the objective function is non- decreasing and there are a finite number of data points. The PCA-L1 procedure converges to a projection vector. 24

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Proof The objective function has a local maximum value at. Because converges to by the PCA-L1 procedure, for all. By Step 4b, for all. 25

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Proof There exists a small neighborhood of, such that if, then for all. Then, since is parallel to, the inequality holds for all. is a local maximum point. 26

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Algorithms So, the PCA-L1 procedure finds a local maximum point. Because is a linear combination of data points, i.e.,, it’s invariant to rotations. Under rotational transformation R:X→RX, then W→RW. 27

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Algorithms Computation complexity: is the number of iterations for convergence. does not depend on the dimension. 28

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Algorithms The PCA-L1 procedure just finds a local maximum solution. It may not be the global solution. We can set appropriately. – By setting. – Run the PCA-L1 with different initial vector. 29

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Algorithms Extracting Multiple Features : 30 Original PCA’s thought. Run the PCA-L1 for each feature dimension.

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Algorithms How to guarantee the orthonormality of the projection vectors? We should show that is orthogonal to. 31

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Proof The projection vector is a linear combination of samples. It’s in the subspace spanned by. Then, we consider : 32 Form Greedy search algorithm. normal vector, (=1)

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Proof Because, is orthogonal to.. is orthogonal to. 33 The orthonormality of the projection vectors is guaranteed.

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Algorithms Even if the greedy search algorithm does not provide the optimal solution, it provides a set of good projections that maximize L1 dispersion. 34

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Algorithms For data analysis, we could decide how much data could be captured. In PCA, we could compute the eigenvalue: 35 The eigenvalue is equivalent to the variance of the feature. We can compute the ratio of the variance to the total variance. The sum of variance: In eigenvalue, it exceeds 95% of the total variance, m is set to.

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Algorithms In PCA-L1, once is obtained, we can compute the variance of the feature. The sum of variance: The total variance: 36 We can set the appropriate number of extracted features like original PCA.

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Experiments In the experiments, we apply PCA-L1 algorithm and compared with R1-PCA and original PCA. Three experiments: – A Toy problem with an Outlier – UCI Data Sets – Face Reconstruction 37

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A Toy Problem with an Outlier Consider the data points in a 2D space: If we discard the outlier, the projection vector should be. 38 an outlier.

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A Toy Problem with an Outlier The projection vector: 39 outlier

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A Toy Problem with an Outlier The residual error : 40 outlier Average residual error PCA-L1L2-PCAR1-PCA 1.2001.4011.206 Much influenced by the outlier.

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UCI Data Sets Data sets in UCI machine learning repositories. Compare the classification performances. 1-NN classifier was used and 10-fold cross validation for average classification rate. For PCA-L1, we set the initial projection vector as. 41

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UCI Data Sets The data sets: 42

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UCI Data Sets The average correct classification rates: 43

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UCI Data Sets The average correct classification rates: 44

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UCI Data Sets The average correct classification rates: 45 In many cases, PCA-L1 outperformed L2-PCA and R1- PCA when the number of extracted features was small.

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UCI Data Sets Average Classification rate on UCI Data Sets: 46 PCA-L1 outperformed other methods by 1% on average.

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UCI Data Sets Computation cost: 47

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Face Reconstruction The Yale face database. – 11 individuals. – 15 face images for one person. Among 165 images, 20% were selected randomly and occluded with a noise block. 48

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Face Reconstruction For these image sets, we applied: – L2-PCA(eigenface) – R1-PCA – PCA-L1 Then, we used extracted features to reconstruct images. 49

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Face Reconstruction 50 Experimental results:

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Face Reconstruction The average reconstruction error is: 51 original image reconstructed image Form 10~20 features, the difference became apparent and PCA-L1 outperformed than other methods.

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Face Reconstruction We added 30 dummy images consist of random black and white dots to the original 165 Yale images. We applied: – L2-PCA(eigenface) – R1-PCA – PCA-L1 We reconstructed images with features. 52

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Face Reconstruction Experimental results: 53

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Face Reconstruction The average reconstruction error: 54 From 6 to 36 features, the error of L2- PCA is constant. The dummy images serious affect those projection vectors. From 14 to 36 features, the error of R1- PCA is increasing. The dummy images serious affect those projection vectors.

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Conclusion The PCA-L1 was proven to find a local maximum point. The computation complexity is proportional to – the number of samples – the dimension of input space – The number of iterations The method is usually faster and robust to outliers. 55

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Principal Component Analysis Given a dataset of l samples: We represent D by projecting the data onto a line running through the sample mean, denoted as ( ): 56

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Principal Component Analysis Then, 57

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Principal Component Analysis To look for the best direction, 58 scatter matrix

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Principal Component Analysis We want to minimize : Maximize, subject to We use Lagrange multipliers: 59

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Principal Component Analysis Since, minimizing can be achieved by choosing as the largest eigenvector of. Similarly, we can extend 1-d to -d projection. 60

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