Non-linear Decision Boundary

Definition

Non-linear Decision Boundary

A non-linear decision boundary is a boundary between class regions that is not an affine hyperplane. For a binary classifier with a continuous score that changes sign across its zero set, the boundary is

The boundary is non-linear if no and satisfy

Unlike a linear decision boundary, need not be flat; it may be curved or have several components.

Through a feature map

A non-linear feature map can produce a non-linear boundary from a linear boundary in feature space:

The non-linearity of alone does not guarantee that is non-linear; the geometry depends on both and the chosen hyperplane .

Example

A circular boundary

For , the boundary is

Negative scores occur inside the unit circle , and positive scores occur outside it. The feature map turns the same score into , whose boundary in feature space is a straight line.

A non-linear score with a linear boundary

The score is non-linear, but its zero set is the line . A non-linear score does not necessarily give a non-linear decision boundary.