[MINI] Feed Forward Neural Networks

Feed Forward Neural Networks In a feed forward neural network, neurons cannot form a cycle. In this episode, we explore how such a network would be able to represent three common logical operators: OR, AND, and XOR. The XOR operation is the interesting case. Below are the truth tables that describe each of these functions. AND Truth Table Input 1 Input 2 Output 0 0 0 0 1 0 1 0 0 1 1 1 OR Truth Table Input 1 Input 2 Output 0 0 0 0 1 1 1 0 1 1 1 1 XOR Truth Table Input 1 Input 2 Output 0 0 0 0 1 1 1 0 1 1 1 0 The AND and OR functions should seem very intuitive. Exclusive or (XOR) if true if and only if exactly single input is 1. Could a neural network learn these mathematical functions? Let's consider the perceptron described below. First we see the visual representation, then the Activation function , followed by the formula for calculating the output.         Can this perceptron learn the AND function? Sure. Let and What about OR? Yup. Let and An infinite number of possible solutions exist, I just picked values that hopefully seem intuitive. This is also a good example of why the bias term is important. Without it, the AND function could not be represented. How about XOR? No. It is not possible to represent XOR with a single layer. It requires two layers. The image below shows how it could be done with two laters.     In the above example, the weights computed for the middle hidden node capture the essence of why this works. This node activates when recieving two positive inputs, thus contributing a heavy penalty to be summed by the output node. If a single input is 1, this node will not activate. Universal approximation theorem tells us that any continuous function can be tightly approximated using a neural network with only a single hidden layer and a finite number of neurons. With this in mind,