The kernel trick (RBF & Polynomial)

One class rings the origin; the other rings that ring. Drag the line anywhere — there's always at least one point on the wrong side. This isn't a bad line; it's that no line can do better than about two-thirds here, because a straight cut can't tell "close to the center" from "far from it."

The data isn't separable in $(x, y)$ — but it doesn't have to stay in $(x, y)$. Lift every point through one new feature, the squared distance from the origin:

• $z$ — the new lifted feature: a point's squared distance from the origin.
• $x$, $y$ — the point's original two coordinates.

1. Radius collapses points regardless of angle

   • Points near the center now share one small value of $z$, no matter which direction they're in.
   • Points on the outer ring share one large value, no matter which direction they're in.
2. Angle was the confusion, radius was the signal

   The two classes were tangled by angle but separated by radius all along — $z$ just throws away the angle and keeps the radius.

This is the exact same 16 points, plotted by $z$ instead of $(x,y)$. One threshold now separates them completely — it's Chapter 7's split all over again, just on a feature that didn't exist until this chapter lifted it into being.

1. Transform an inner and an outer point

   • $(1.8, 0)$: $z = 1.8^2 = 3.24$.
   • $(4.7, 0)$: $z = 4.7^2 = 22.09$.
2. Transform a diagonal point at the same inner radius

   $(1.556, 1.556)$: $z = 1.556^2 + 1.556^2 \approx 2.42+2.42 = 4.84$ — the same value every inner point at that radius gets, regardless of angle.

3. Confirm a single threshold separates everything

   • Every inner point lands with $z \le 4.84$.
   • Every outer point lands with $z \ge 22.09$.

   Any threshold in between — 10, 15, 20 — separates all sixteen points at once.