Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

Neural networks 261

But this last sum over the K compone nts of the label ℓ(ν) is just unity, and

therefore we have

−δ

o

k

(ν) = −ℓ

k

(ν) + m

k

(ν), k = 1 . . . K,

which may be written as the K-component vector

δ

o

(ν) = ℓ(ν) − m(ν). (6.38)

From Equation (6.36), we can ther efo re express the third step in the back-

propagation algorithm in the form of the matrix equatio n (see Exercise 6)

W

o

(ν + 1) = W

o

(ν) + η n(ν)δ

o

(ν)

⊤

. (6.39)

Here W

o

(ν + 1) indicates the synaptic weight matrix after the up date for νth

training pair. Note that the second term on the right-hand side of Equation

(6.39) is an o uter product, yielding a matrix of dimension (L + 1) ×K and so

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