How to design a neural network to predict the quadrant where a given point lies?

Question

I've written a single perceptron that can predict whether a point is above or below a straight-line graph, given the correct training data and using a sign activation function.

Now, I'm trying to design a neural network that can predict whether a point $(x, y)$ is in the 1st, 2nd, 3rd or 4th quadrant of a graph.

One idea I have had is to have 2 input neurons, the first taking the $x$ value, the 2nd taking the $y$ value, these then try and predict individually whether the answer is on the right or left of the centre, and then above or below respectively. These then pass their outputs to the 3rd and final output neuron. The 3rd neuron uses the inputs to try and predict which quadrant the coordinates are in. The first two inputs use the sign function.

The problem I'm having with this is to do with the activation function of the final neuron. One idea was to have a function that somehow scaled the output into an integer between 0 and 1, so 0 to 0.25 would be quadrant 1, and so on up to 1. Another idea would be to convert it to a value using sin and representing it as a sine wave as this could potentially represent all 4 quadrants.

Another idea would be to have a single neuron taking the input of the $x$ and $y$ value and predicting whether something was above or below a graph (like my perceptron example), then having two output neurons, which the 1st output neuron would be fired if it was above the line and then passed in the original $x$ coordinate to that output neuron. The 2nd output neuron would be fired if it was below, then pass in the original $x$ value, as well to determine if it was left or right.

Are these good ways of designing a neural network for this task?

Answer 1

First of all, what you are trying to do can be achieved by simple logical programming. Secondly, you are making things overly complex.

$1$ node in a neural network can predict as many values as you would like it to, as it outputs a real number no matter the activation function. It is us who round of the number to $0$ or $1$ or maybe $0, 0.5, 1$ depending on the classification job at hand.

$2$ neurons can predict $4$ classes $00, 01, 10, 11$ depending on how you use it.

Now, in your specific problem, 2 neurons are sufficient, since not only can you decompose the problem into two separate problems, both the problems are linearly separable too. This is not always the case or it may be too complex to find linearly separable classes, so, in general, one output node is reserved for $1$ class only. If it is that class only that output node is activated while the rest is $0$. Memory and process consuming, but easier.

Coming to your first approach. I don't exactly know how to solve a 4-class classification problem with a single output node. A minimum of 2 nodes is required. The problem with your approach is that, if you think critically, your hidden nodes are not doing anything other than scaling the inputs, so almost the same input is propagated and then you are asking a node to solve a 4-class classification problem, which is not linearly separable (I think). $\sin$ will not be able to separate it if you think critically.

The second idea is better. Just remove the hidden node and you will be able to classify it based on $00, 01, 10, 11$ outputs.

Although this problem can be solved by simple logical computation and your neural network is basically doing nothing.