A LEAST SQUARES FORMULATION FOR LDA

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In this section, we discuss recent developments on connecting LDA to multivariate linear regression (MLR). We rst discuss the relationship between linear regression and LDA in the binary-class case. We then present multivariate linear regression with a speci c class indicator matrix. This indicator matrix plays a key role in establishing the equivalence relationship between MLR and LDA in the multiclass case.

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Linear Regression versus Fisher LDA

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Given a data set of two classes, {(xi , yi )}n , xi IRd and yi { 1, 1}, the linear i=1 regression model with the class label as the output has the following form: f (x) = xT w + b, (1.16)

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where w IRd is the weight vector, and b is the bias of the linear model. A popular approach for estimating w and b is to minimize the sum-of-squares error function, called least squares, as follows: L(w, b) = 1 2

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||f (xi ) yi ||2 =

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1 T ||X w + be y||2 , 2

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(1.17)

where X = [x1 , x2 , . . . , xn ] is the data matrix, e is the vector of all ones, and y is the vector of class labels. Assume that both {xi } and {yi } have been centered, that is,

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1.4 A Least Squares Formulation for LDA

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n i=1 xi

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= 0 and

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n i=1 yi

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= 0. It follows that yi { 2n2 /n, 2n1 /n} ,

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where n1 and n2 denote the number of samples from the negative and positive classes, respectively. In this case, the bias term b in Eq. (1.16) becomes zero and we construct a linear model f (x) = xT w by minimizing 1 T ||X w y||2 . (1.18) 2 It can be shown that the optimal w minimizing the objective function in Eq. (1.18) is given by [16, 17] L(w) = w = XXT

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Note that the data matrix X has been centered and thus XXT = nSt and Xy = 2n1 n2 (1) c(2) ). It follows that n (c w= 2n1 n2 F 2n1 n2 + (1) St (c c(2) ) = G , 2 n n2

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where GF is the optimal solution to FLDA in Eq. (1.14). Hence linear regression with the class label as the output is equivalent to Fisher LDA, as the projection in FLDA is invariant of scaling. More details on this equivalence relationship can be found in references 15, 16, and 35.

1.4.2 Relationship Between Multivariate Linear Regression and LDA

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In the multiclass case, we are given a data set consisting of n samples {(xi , yi )}n , i=1 where xi IRd and yi {1, 2, . . . , k} denotes the class label of the ith sample and k > 2. To apply the least squares formalism to the multiclass case, the 1-of-k binary coding scheme is usually used to associate a vector-valued class code to each data point [15, 17]. In this coding scheme, the class indicator matrix, denoted as Y1 IRn k , is de ned as follows: Y1 (ij) = 1 0 if yi = j, otherwise. (1.19)

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It is known that the solution to least squares problem approximates the conditional expectation of the target values given the input [15]. One justi cation for using the 1-of-k scheme is that, under this coding scheme, the conditional expectation is given by the vector of posterior class probabilities. However, these probabilities are usually approximated rather poorly [15]. There are also some other class indicator matrices considered in the literature. In particular, the indicator matrix Y2 IRn k , de ned as Y2 (ij) = 1 1/(k 1) if yi = j, otherwise, (1.20)

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1

Discriminant Analysis for Dimensionality Reduction

has been introduced to extend support vector machines (SVM) for multiclass classi cation [36] and to generalize the kernel target alignment measure [37], originally proposed in reference 38. In multivariate linear regression, a k-tuple of discriminant functions f (x) = (f1 (x), f2 (x), . . . , fk (x)) is considered for each x IRd . Denote X = [ 1 , . . . , xn ] IRd n and Y = Yij x n k as the centered data matrix X and the centered indicator matrix Y , respectively. IR 1 1 That is, xi = xi x and Yij = Yij Yj , where x = n n xi and Yj = n n Yij . i=1 i=1 k d , of the k linear models, Then MLR computes the weight vectors, {wj }j=1 IR fj (x) = xT wj , for j = 1, . . . , k, via the minimization of the following sum-of-squares error function: L(W) = 1 1 T F ||X W Y ||2 = 2 2