Consider a Multi$-$layer Perceptron $($MLP$)$ for the following two general tasks: $(1)$

multi$-$class classification of $K=5$ categories with $5$ output units; and $(2)$ regression with a single

output unit, where each hidden unit in both tasks uses a hyperbolic tangent function such that

$z_h^t=tanh({\displaystyle \underset{j=1}{\overset{D}{}}}{w}_{hj}{x}_j^t+w_{h0}).$ The output unit in classification uses a softmax activation function

such that $y_i^t=\frac{\text{e}\text{x}\text{p}({\displaystyle \underset{h}{\overset{?}{}}}{v}_{ih}{z}_h^t+v_{i0})}{{\displaystyle \underset{j}{\overset{?}{}}}\text{e}\text{x}\text{p}({\displaystyle \underset{h}{\overset{?}{}}}{v}_{jh}{z}_h^t+v_{j0})}.$ The error functions for tasks $(1)$ and $(2)$ are given below

respectively:

Multi$-$class classification: $E(W,V|x)=-{\displaystyle \underset{t=1}{\overset{N}{}}}{\displaystyle \underset{i=1}{\overset{K}{}}}{r}_i^{t}log{y}_i^t+\frac{}{2}{\displaystyle \underset{h=1}{\overset{H}{}}}||w_h|{|}_2^2+\frac{}{2}{\displaystyle \underset{k=1}{\overset{K}{}}}||v_k|{|}_2^2$

Regression: $E(W,v|x)=\frac{1}{2}{\displaystyle \underset{t=1}{\overset{N}{}}}(r^t-y^t{)}^2+\frac{}{2}{\displaystyle \underset{h=1}{\overset{H}{}}}||w_h|{|}_2^2+\frac{}{2}||v|{|}_2^2.$

$($a$)$ Draw two Multi$-$layer Perceptrons, each for one of the above tasks, showing: input values

$x_{0}dots{x}_D,$ output of the hidden units $z_{0}dots{z}_H,$ weights $W$ and $V($or $v),$ and the output$($s$)($i$.$e$.,$

$y_i$ of output unit i for multi$-$class classification, and $y$ for regression$).$ Note the difference in

the structure between the two tasks $($you may write or draw$).$