XY3 = (nSt )+ nHb = St+ Hb .

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(1.25)

It can be shown that the transformation matrix GLDA of LDA, which consists of the top eigenvectors of St+ Sb , and the projection matrix for MLR that is given in Eq. (1.25) are related as follows [40]: W MLR = GLDA , 0 QT , where is a diagonal matrix and Q is an orthogonal matrix. The K-Nearest-Neighbor (K-NN) algorithm [16] based on the Euclidean distance is commonly applied as the classi er in the dimensionality-reduced space of LDA. If we apply W MLR for dimensionality reduction before K-NN, the matrix W MLR is invariant of an orthogonal transformation, since any orthogonal transformation preserves all pairwise distance. Thus W MLR is essentially equivalent to GLDA , 0 or GLDA , as the removal of zero columns does not change the pairwise distance either. Thus the essential difference between W MLR and GLDA is the diagonal matrix . Interestingly, it was shown in reference 40 that the matrix is an identity matrix under the condition C1 de ned in Eq. (1.15). This implies that multivariate linear regression with Y3 as the class indicator matrix is equivalent to LDA provided that

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1

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Discriminant Analysis for Dimensionality Reduction

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the condition C1 is satis ed. Thus LDA can be formulated as a least squares problem in the multiclass case. Experimental results in reference 40 show that condition C1 is likely to hold for high-dimensional and undersampled data.

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SEMISUPERVISED LDA

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Semisupervised learning, which occupies the middle ground between supervised learning (in which all training examples are labeled) and unsupervised learning (in which no labeled data are given), has received considerable attention recently [41 43]. The least square LDA formulation from the last section results in Laplacianregularized LDA [44]. Furthermore, it naturally leads to semisupervised dimensionality reduction by incorporating the unlabeled data through the graph Laplacian.

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Graph Laplacian

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Given a data set {xi }n , a weighted graph can be constructed where each node in the i=1 graph corresponds to a data point in the data set. The weight Sij between two nodes xi and xj is commonly de ned as follows: exp Sij = 0

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xi xj

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xi N (xj ) or xj N (xi ), otherwise,

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(1.26)

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where both and > 0 are parameters to be speci ed, and xi N (xj ) implies that xi is among the nearest neighbors of xj [45]. Let S be the similarity matrix whose (i, j)th entry is Sij . To learn an appropriate representation {zi }n which preserves i=1 locality structure, it is common to minimize the following objective function [45]: zi z j

i,j 2

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Sij .

(1.27)

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Intuitively, if xi and xj are close to each other in the original space that is, Sij is large then zi zj tends to be small if the objective function in Eq. (1.27) is minimized. Thus the locality structure in the original space is preserved. De ne the Laplacian matrix L as L = D S, where D is a diagonal matrix whose diagonal entries are the column sums of S. That is, Dii = n Sij . Note that j=1 L is symmetric and positive semide nite. It can be veri ed that 1 2

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zi z j

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i=1 j=1

Sij = trace(ZLZT ),

(1.28)

where Z = [z1 , . . . , zn ].

1.6 Extensions to Kernel-induced Feature Space

1.5.2 A Regularization Framework for Semisupervised LDA

In semisupervised LDA, information from unlabeled data is incorporated into the formulation via a regularization term de ned as in Eq. (1.28). Mathematically, semisupervised LDA computes an optimal weight matrix W , which solves the following optimization problem: W = arg min

X T W Y3

+ trace(W T XLXT W) ,

(1.29)

where 0 is a tuning parameter and Y3 is the class indicator matrix de ned in Eq. (1.24). Since the Laplacian regularizer in Eq. (1.29) does not depend on the label information, the unlabeled data can be readily incorporated into the formulation. Thus the locality structures of both labeled and unlabeled data points are captured through the transformation W. It is clear that W is given by W = XLXT + XXT