$1)$ Computing Network Sizes

In this question, you will analyze the size of a state$-$of$-$the$-$art architecture discussed in class. In the ImageNet ILSVRC $2014$ contest, two fairly deep networks performed very well: the VGG network and the GoogLeNet. While VGG performed best in the localization task and ranked second in classification, GoogLeNet won the classification task achieving a top$-5$ error rate of $6\mathrm{.}67\%.$ Figure $1$ presents the architecture of GoogLeNet, which is built up of $9$ stacked $$inception$$ modules displayed in Figure $2.$

$($a$)$ Consider the layer $$inception $(3$a$)$ from Table $1($Figure $3)$ in the GoogLeNet paper. Notice how $33$ reduce$$ and $55$ reduce$$ are used between layers: according to Section $5$ of the paper, this $$stands for the number of $1$x$1$ filters in the reduction layer used before the $33$ and $55$ convolutions$.$ Compute the number of free parameters for the $$inception $(3$a$)$ layer.

$($b$)$ Now, suppose that the reduction portion of $33$ reduce$$ and $55$ reduce$$ were omit$-$ ted $($i$.$e$.,$ no $11$ filters were used before the $33$ and $55$ convolutions$).$ Compute the number of free parameters for this updated $$inception $(3$a$)$ layer. How does this compare to the original layer?