2. The index (3.1.8) becomes E(W, b) = =1 n i=1(yi (Wx )i bi...
Question:
2. The index (3.1.8) becomes E(W,
b) =
κ=1
n i=1(yκi − (Wxκ )i − bi )2, where yκi is the ith component of the κth target yκ . In order to find normal equations, define
ˆW
=
⎛
⎜⎜⎜⎜⎜⎝
w11 w12 . . . w1n w21 w22 . . . w2n
......
wd1 wd2 . . . wdn b1 b2 . . . bn
⎞
⎟⎟⎟⎟⎟⎠
, and let (Y )ij = yij with Y ∈ R
,n. Then E( ˆ W) = tr
(Y − ˆX
ˆW
)(Y
− ˆW
ˆX
)
=
tr(YY
) − 2 tr(Y ˆW
ˆX
) + tr( ˆX
ˆW
ˆW
ˆX
), and the normal equations are obtained imposing
∂ ˆW E = 0. Since ∂ ˆW tr(Y ˆW
ˆX
) = ˆX Y and ∂ ˆW tr( ˆX
ˆW
ˆW
ˆX
) = 2( ˆX
ˆX ˆ W),
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