Consider the following code lines $($DONT USE AI$)$:

import numpy as np

import matplotlib.pyplot as plt

from tensorflow import keras

from tensorflow.keras import layers

from tensorflow.keras.datasets import mnist

from tensorflow.keras.models import Sequential

$($x$\_$train, y$\_$train$),($x$\_$test, y$\_$test$)=$ mnist.load$\_$data$()$

x$\_$train $=$ x$\_$train.astype$("$float$32")/255\mathrm{.}0$

x$\_$test $=$ x$\_$test.astype$("$float$32")/255\mathrm{.}0$

x$\_$train $=$ np$.$expand$\_$dims$($x$\_$train, $-1)$

x$\_$test $=$ np$.$expand$\_$dims$($x$\_$test, $-1)$

input$\_$shape $=$ x$\_$train.shape$[1$:$]$

num$\_$classes $=10$

epochs $=1000$

def build$\_$mlp$\_$model$()$:

model $=$ keras.Sequential$([$

layers.Flatten$($input$\_$shape$=$input$\_$shape$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$(4,$ activation$=$"relu"$),$

layers.Dense$($np$.$prod$($input$\_$shape$),$ activation$=$"sigmoid"$),$

layers.Reshape$($input$\_$shape$)$

$])$

return model

mlp$\_$model $=$ build$\_$mlp$\_$model$()$

mlp$\_$model.compile$($optimizer$="$sgd$",$ loss$=$"mse"$)$

mlp$\_$model.summary$()$

mlp$\_$history $=$ mlp$\_$model.fit$($x$\_$train$[$:$1],$ x$\_$train$[$:$1],$ epochs$=$epochs, batch$\_$size$=1,$ verbose$=1)$

plt$.$plot$($mlp$\_$history.history$[\text{'}$loss$\text{'}])$

plt$.$title$(\text{'}$MLP Training Loss'$)$

plt$.$xlabel$(\text{'}$Epoch$\text{'})$

plt$.$ylabel$(\text{'}$Loss$\text{'})$

plt$.$show$()$

def build$\_$cnn$\_$model$()$:

model $=$ keras.Sequential$([$

layers.Conv$2$D$(10,$ kernel$\_$size$=(5,5),$ activation$=$'relu', padding$=$'same', input$\_$shape$=$input$\_$shape$),$

layers.Conv$2$D$($input$\_$shape$[-1],$ kernel$\_$size$=(1,1),$ activation$=$'sigmoid', padding$=$'same'$)$

$])$

return model

cnn$\_$model $=$ build$\_$cnn$\_$model$()$

cnn$\_$model.compile$($optimizer$="$sgd$",$ loss$=$"mse"$)$

cnn$\_$model.summary$()$

cnn$\_$history $=$ cnn$\_$model.fit$($x$\_$train$[$:$1],$ x$\_$train$[$:$1],$ epochs$=$epochs, batch$\_$size$=1,$ verbose$=1)$

plt$.$plot$($cnn$\_$history.history$[\text{'}$loss$\text{'}])$

plt$.$title$(\text{'}$CNN Training Loss'$)$

plt$.$xlabel$(\text{'}$Epoch$\text{'})$

plt$.$ylabel$(\text{'}$Loss$\text{'})$

plt$.$show$()$

Answer these questions:

Question$1-$ Consider a multilayer ReLU network h: R$^$n $($up to$)$ R$^$n such that h$($x$)=$ W$3$ReLU$($W$2$ReLU$($W$1$x$+$b$1)+$b$2)+$b$3$ with W$1($as an element of$)$R$^$a x n$,$ W$2($as an element of$)$ R$^$ n x a$,$ W$3($as an element of$)$ R$^$n x n$,$ b$1($as element of $)$R$^$a; b$2,$ b$3($as element of $)$ R$^$n$.$ Find a possible solution for W$1,$ W$2,$ W$3,$ b$1,$ b$2,$ b$3$ such that h represents the identity function.

What if you want h to represent a constant function that always outputs x$0?$

Question$2$:

P$3\mathrm{.}2-$ Consider a CNN g: R$^$n x n $($up to$)$ R$^$n x n model composed by a first hidden convolutional layer with c filters, d x d $($d$>1$ odd$)$ kernel, identity activation and a suitable convolutional output layer. Find a possible architecture for g $($i$.$e$.$ specify the complete architecture, c$,$ the values in the filters, padding and stride$)$ such that g represents the identity function.

If instead of the identity activation, we use a ReLU activation, how should the architecture change?

$($Note: R for set of real numbers$)$

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