Pattern Recognition and Machine Learning

5.4. The Hessian Matrix 255

usual by summing over the contributions from each of the patterns separately. For
the two-layer network, the forward-propagation equations are given by

aj =

∑

i

wjixi (5.98)

zj = h(aj) (5.99)

yk =

∑

j

wkjzj. (5.100)

We now act on these equations using theR{·}operator to obtain a set of forward
propagation equations in the form

R{aj} =

∑

i

vjixi (5.101)

R{zj} = h′(aj)R{aj} (5.102)

R{yk} =

∑

j

wkjR{zj}+

∑

j

vkjzj (5.103)

wherevjiis the element of the vectorvthat corresponds to the weightwji. Quan-
tities of the formR{zj},R{aj}andR{yk}are to be regarded as new variables
whose values are found using the above equations.
Because we are considering a sum-of-squares error function, we have the fol-
lowing standard backpropagation expressions:

δk = yk−tk (5.104)

δj = h′(aj)

∑

k

wkjδk. (5.105)

Again, we act on these equations with theR{·}operator to obtain a set of backprop-
agation equations in the form

R{δk} = R{yk} (5.106)

R{δj} = h′′(aj)R{aj}

∑

k

wkjδk

+h′(aj)

∑

k

vkjδk+h′(aj)

∑

k

wkjR{δk}. (5.107)

Finally, we have the usual equations for the first derivatives of the error

∂E

∂wkj

= δkzj (5.108)

∂E

∂wji

= δjxi (5.109)