# What exactly happens when I do a feature cross?

I was going through a machine learning course and they talked about combining various features to create synthetic feature to take care of non linear data. For eg in the below picture I didn't do any feature crossing and the model didn't fit:

But if I do some feature crossing and create/activate features $$x_1^2$$, $$x_2^2$$ and $$x_1x_2$$ I get this:

The model fits now. But why? What exactly does feature crossing do that enables a model to fit non linear data?

But it seems like it actually is separable with a circle/ellipse (let's say it's inside a circle to simplify the problem): it seems reasonable to have hypothesis that, for some $c$ if $x^2 + y^2< c$ then a point is blue.
That means that if you use $x^2, y^2$ as features, you can fit a linear classifier to these data points and actually separate the classes linearly.
• If you take just $x^2, y^2$ then this is correct reasoning. But I don't know if it works exactly that way in this simulation - I don't see if you disabled the original features. Mar 4, 2018 at 10:48
• Ok, I just tried it in the simulation with original features disabled and only $x_1^2$, $x_2^2$ & $x_1x_2$ enabled. It still fits, better in fact. So should I conclude that original features do not matter now? Why do they not matter anymore? Is it because the information they used to carry are redundant now? Mar 4, 2018 at 13:11