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By Xu B.G.

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Extra info for A 3-color Theorem on Plane Graphs without 5-circuits

Example text

K, of each class are computed. The within-class scatter is defined: Sw = mj ∑kj=1 p jC( j) , where p j = m . The between-class scatter is defined: Sb = C − Sw . 2. The eigenvectors and eigenvalues of the matrix S = Sw−1 Sb are computed. The eigenvectors are sorted in descending order of the corresponding eigenvalues. d eigenvectors corresponding to the highest eigenvalues are selected (under the requirement that d < k). ¯ d , i = 1, . . , m, is performed, where Ad is a 3. The transformation Yi = (Xi − X)A d-column matrix consisting of the eigenvectors corresponding to the highest d eigenvalues of matrix S.

The minimization problem is high dimensional: the number of variables is N = md. • The function is not everywhere differentiable. • It is invariant with respect to translation, rotation, and mirroring. Some of the mentioned difficulties could be avoided at least partially. It is possible to avoid invariance when introducing constraints or fixing some of the coordinates of image points. The invariance with respect to translation can be avoided by centering the image requiring that the sum of coordinate values of image points be zero: m ∑ yik = 0, k = 1, .

The diagonal elements rkk , k = 1, . . , n, are equal to 1. This matrix is symmetric. 2 Dimensionality Reduction 23 ckl = 1 m ∑ (xik − x¯k )(xil − x¯l ). e. ckk is the variance of feature xk . The covariance matrix C consists of the covariance coefficients: C = {ckl , k, l = 1, . . , n}. 3) that the correlation coefficient is equal to ckl rkl = √ . 5) If the features xk and xl are not correlated, their covariance coefficient is equal to zero: ckl = clk = 0, k = l. Let us describe the eigenvector and the eigenvalue of the covariance matrix.

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