Here are my solutions to exercise 2.
There is a way of determining the bitwise representation of a digit by adding an extra layer to the three-layer network above. The extra layer converts the output from the previous layer into a binary representation, as illustrated in the figure below. Find a set of weights and biases for the new output layer. Assume that the first layers of neurons are such that the correct output in the third layer (i.e., the old output layer) has activation at least , and incorrect outputs have activation less than .
This is converting a base-ten value to base-two value.
I first listed out all the possible numbers.
I noticed that the first thing I could do it solve for the first binary digit. The first binary digit tells us whether or not the number is odd or even. I found that the bias is unnecessary since the dot product will produce either or . The weights for the first node are as follows:
I am going to denote the weight due to number in the old output layer, ,
So if the value is odd the and if even .
For node two it should only be activated if the numbers are
For node three it should only be activated if the numbers are
For node four it should only be activated if the numbers are
Again, there are no biases for any of the nodes.