Suppose that you want to upload data to the cloud for some mysterious purpose. You want to limit your network usage; so you want to optimise every bit of data that you are going to upload. The data is a collection of m$-$dimensional data points. Take an example: for m $=2$ you have data X $=\{(1,2),(1\mathrm{.}5,3),(2,4),(6,12),(2\mathrm{.}5,5)\}.$

You can clearly see that the second feature of each data point is just twice the first feature. So$,$ instead of uploading both the features for each data points, you can just upload the first feature $($or second feature$)$ and the second feature $($or first feature$)$ can be computed at the cloud end. Of course, such a compression mechanism is lossy; but, it would save the network traffic by $50\%.$ The problem becomes difficult when data points are higher in dimension $($m $>=3)$ or number of data points is large. As there is no way of having a complete feature$-$level relationship. Any assumptions related to this would be unrealistic. Therefore, you want to use neural network for the purpose of data compression.

For your answers, use the following notations:

$$ m$-$dimensional data point, x $=($x$1,$ x$2,...,$ xm$)$ and x in X

$$ n$-$dimensional compressed data point z $=($z$1,$ z$2,...,$ zn$)$

$$ Loss function L

$$ Define additional notations before using them.

Answer the following questions:

a$)$ Write the complete procedural steps for uploading and downloading data.

b$)$ Draw the architecture of your neural net.

c$)$ Write all the equations that are needed to design the system.

d$)$ Write the loss function that the neural network should optimize.