Watching a nonlinear problem become linear

The red points sit inside a blue ring, so no straight line in the original plane can separate the classes. Lift each point with a quadratic feature map, z = x² + y², and the same data becomes separable by a flat plane in 3D.

Data concentric classes

Lift quadratic feature map

Separator line in 2D, plane in 3D

View

2D plane

full 2d full 3d

No line works in the original plane

The red class is surrounded by the blue class. Any straight separator leaves points from both classes mixed on at least one side.

Inner class

Outer class

Separating plane after lifting

Feature map: (x, y) -> (x, y, x² + y²)

At full 3D, the camera reaches a side view, so the lifted bowl reads as an upward-opening parabola. The green separator becomes a horizontal cut through that surface, corresponding to a curved boundary back in the original plane.