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\begin{document}

\title{Introduction to Machine Learning\\
Homework 8:  Convolutional Neural Networks}
\author{Prof. Sundeep Rangan}
\date{}

\maketitle

Submit answers to only problems 1--3.  But, make sure you know how to do all problems.

\begin{enumerate}

\item Let $X$ and $W$ be arrays,
\[
    X = \left[ \begin{array}{ccccc}
        0 & 0 & 0 & 0 & 0 \\
        0 & 3 & 3 & 3 & 0 \\
        0 & 3 & 3 & 3 & 0 \\
        0 & 3 & 2 & 3 & 0 \\
        0 & 3 & 2 & 3 & 0 \\
        0 & 0 & 0 & 0 & 0
        \end{array} \right], \quad
    W = \left[ \begin{array}{cc}
        1 & -1 \\
        1 & -1
        \end{array} \right].
\]
Let $Z$ be the 2D convolution (without reversal):
\beq \label{eq:Zconv}
    Z[i,j] = \sum_{k_1,k_2} W[k_1,k_2]X[i+k_1,j+k_2].
\eeq
Assume that the arrays are indexed starting at $(0,0)$.
\begin{enumerate}[(a)]
\item What are the limits of the summations over $k_1$ and $k_2$ in \eqref{eq:Zconv}?
\item What is the size of the output $Z[i,j]$ if the convolution is computed only on the \emph{valid} pixels (i.e.\ the pixel locations $(i,j)$
where the summation in \eqref{eq:Zconv} does not exceed the boundaries of $W$ or $X$).
\item What is the largest positive value of $Z[i,j]$ and state one pixel location $(i,j)$ where that value occurs.
\item What is the largest negative value of $Z[i,j]$ and state one pixel location $(i,j)$ where that value occurs.
\item Find one pixel location where $Z[i,j]=0$.
\end{enumerate}

\item Suppose that a convolutional layer of a neural network has an input tensor $X[i,j,k]$ and computes
an output via a convolution and ReLU activation,
\begin{align*}
    Z[i,j,m] &= \sum_{k_1} \sum_{k_2} \sum_n W[k_1,k_2,n,m]X[i+k_1,j+k_2,n] + b[m], \\
    U[i,j,m] &= \max\{0, Z[i,j,m] \}.
\end{align*}
for some weight kernel $W[k_1,k_2,n,m]$ and bias $b[m]$.  Suppose that $X$ has shape (48,64,10) and $W$ has shape (3,3,10,20).
Assume the convolution is computed on the \emph{valid} pixels.
\begin{enumerate}[(a)]
\item What are the shapes of $Z$ and $U$?
\item What are the number of input channels and output channels?
\item How many multiplications must be performed to compute the convolution in that layer?
\item If $W$ and $b$ are to be learned, what are the total number of trainable parameters in the layer?
\end{enumerate}

\item Suppose that a convolutional layer in some neural network
is described as a linear convolution followed by a sigmoid activation,
\begin{align*}
    Z[i,j,m] &= \sum_{k_1} \sum_{k_2} \sum_n W[k_1,k_2,n,m]X[i+k_1,j+k_2,n] + b[m], \\
    U[i,j,m] &= 1/(1+\exp(-Z[i,j,m])).
\end{align*}
where $X[i,j,n]$ is the input of the layer and $U[i,j,m]$ is the output.
Suppose that during back-propagation, we have computed the gradient $\partial J/\partial U$ for some loss function $J$.
That is, we have computed  the components $\partial J/\partial U[i,j,m]$.  Show how to compute the following:
\begin{enumerate}[(a)]
\item The gradient components $\partial J/\partial Z[i,j,m]$.
\item The gradient components $\partial J/\partial W[k_1,k_2,n,m]$.
\item The gradient components $\partial J/\partial X[i,j,n]$.
\end{enumerate}

\item 
In the previous problem, we considered a single sample.
Suppose there were a mini-batch of samples.
\begin{enumerate}[(a)]
\item How would you represent $Z$ and $U$ for the mini-batch case?
\item Re-write the equations for $Z$ and $U$.  
\item Re-compute the gradients.
\end{enumerate}
\end{enumerate}

\end{document}