Single Perceptron - Non-linear Evaluating function

Question

In the case of a single perceptron - literature states that it cannot be used for seperating non-linear discriminant cases like the XOR function. This is understandable since the VC-dimension of a line (in 2-D) is 3 and so a single 2-D line cannot discriminate outputs like XOR.

However, my question is why should the evaluating function in the single perceptron be a linear-step function? Clearly if we have a non-linear evaluating function like a sigmoid, this perceptron can discriminate between the 1s and 0s of XOR. So, am I missing something here?

Answer 1

if we have a non-linear evaluating function like a sigmoid, this perceptron can discriminate between the 1s and 0s of XOR

That's not true at all. The criteria for discrimination is not the shape of the line (or hyperplane in higher dimensions), but rather whether the function allows linear separability.

There is no single function that produces a hyperplane capable of separating the points of the XOR function. The curve in the image separates the points, but it is not a function.

To separate the points of XOR, you'll have to use at least two lines (or any other shaped functions). This will require two separate perceptrons. Then, you could use a third perceptron to separate the intermediate results on the basis of sign.