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Could someone please give me a mathematical correct explanation of why a Multilayer Perceptron can solve the XOR problem?

My interpretation of the perceptron is as follows:

A perceptron with two inputs enter image description here and enter image description here has the following linear function and is hence able to solve linear separately problems such as AND and OR.

enter image description here

enter image description here is the basic step function.

The way I think of it is that I substitute the two parts within enter image description here separated by the + sign as enter image description here and enter image description hereI get enter image description here which is a line. By applying the step function I get one of the clusters with respect to the input. Which I interpret as one of the spaces separated by that line.

Because the function of an MLP is still linear, how do I interpret this in a mathematical way and more important: Why is it able to solve the XOR problem when it's still linear? Is it because it is interpolating a polynomial?

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The XOr, or “exclusive or”, the problem is a classic problem in ANN research. It is the main problem of using a neural network to predict the outputs of XOr logic gates given two binary inputs. A XOr function should return a true value if the two inputs are not equal and a false value if they are equal. All the possible inputs and their predicted outputs are shown below:

image

Like all ANNs, the perceptron is composed of a network of units, which are analogous to biological neurons. A unit can receive input from other units. In doing so, it takes the sum of all values received and decides whether it is going to forward a signal on to other units to which it is connected. This is called activation. The activation function helps to reduce the sum of input values to a 1 or a 0 (or a value very close to a 1 or 0) in order to represent activation or lack thereof. Another form of unit, known as a bias unit, always activates, typically sending a hardcoded 1 to all units to which it is connected.

Perceptrons include a single layer of input units including one bias unit and a single output unit (as seen below). Here a bias unit is depicted by a dashed circle, while other units are shown as blue circles. There are two non-bias input units representing the two binary input values for XOr. Any number of input units can be included.

image

For more information regarding this, refer the following link:

https://towardsdatascience.com/perceptrons-logical-functions-and-the-xor-problem-37ca5025790a

If you want to know more about Neural Network visit this Neural Network Tutorial.

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