Hello humans, by now you should know what the heck is a perceptron : a way to find the parameters w and b of a binary linear classifer. Recall that we will be using the following definition of a b.l.c :

With a perceptron we can easily find appropriate values of b and w to compute many boolean functions. For example, we can do it easily with the “OR” function :

The “OR” function has the following truth table :

x1 x2 x1 + x2
1 1 1
1 0 1
0 1 1
0 0 0

If we plot these values, we can see that we can separate them by a line :

OR function

So, we have binary values and they are linearly separable, therefore we can implement this function using a perceptron by setting :

(Note that we could have used other values). Just to make sure it works, let’s manually compute the result :

x1 x2 10*x1 + 10*x2 - 5 H
1 1 15 1
1 0 5 1
0 1 5 1
0 0 -5 0

Ok, nice, but what about the XOR function ?

x1 x2 x1 xor x2
1 1 0
1 0 1
0 1 1
0 0 0

Try to separate these points by a single line…

XOR function

Yep, you can’t. Perceptrons can only classify linearly separable data.

But let’s try to be a little bit smart and go back to the definition of the “Xor” :

The xor function is made of 3 basic operations : one OR, and two ANDs. The two ANDs are really similar, let’s call them “partial xor”.

IMAGE TREE OPERATIONS

If we focus on one of the partial xor, we can write :

This function has the following truth table :

x1 x2 x1 Pxor x2
1 1 0
1 0 1
0 1 0
0 0 0

If we plot it, we can see that we can separate the true value from the false by a line.

partial xOR function

We can use a perceptron, if we set (for example) :

If we want to use the same weight vector to implement the second partial xor, we just need to switch the rows of x1 and x2 in x :

Now that we have a way to represent these 3 operations with perceptrons. We can link them together and build a 2-layer perceptron :

IMAGE multi-layer PERCETRONS with weights

By building a multilayer percetron, we can classify non-linearly separable data.

what’s next

The last 3 articles were just an introduction to the subject of neural networks. They will help us understand what neurons are, how they work, why we use multilayers neural networks.

Let’s gooo