A Feature Map Makes XOR Linear
Key concept: XOR cannot be separated by one line in 2D, but lifting the points with ψ(x) = (x₁, x₂, x₁x₂) makes them separable by a single plane in 3D.
The four XOR corners cannot be split into their two classes by any single straight line — the best a line can do is separate 3 of the 4 points. This demo shows the fix: add one extra, nonlinear feature, x₁x₂, and separate the lifted points with a plane instead of a line.
The left panel shows the original 2D square with the best possible line (3/4 correct); the right panel shows the same four points lifted into 3D (x₁, x₂, x₁x₂) space, orbitable by drag, with a separating plane that gets all 4 correct. Press Lift to animate the points rising and the plane fading in, and watch the boundary in the left panel morph from a single line into a cross (x₁ = ½ and x₂ = ½) — exactly the shape one line can never draw.
What to look for: the score badge flips from 3/4 to 4/4 the moment the extra feature becomes available — one new coordinate is enough to make an “impossible” classification problem solvable by a flat plane.
Interaction is easiest on a laptop/desktop screen. On a small screen, you can also open the demo in its own tab at full size.