Easy

RBF Kernel

Easy

~12 min

code completion

Radial Basis Function (RBF) Kernel

Kernel SVMs implicitly map data to a higher-dimensional feature space using a kernel function — no explicit feature mapping needed.

The most popular kernel is the RBF (Gaussian) kernel:

K (x_{1}, x_{2}) = exp (- γ ∥ x_{1} - x_{2} ∥^{2})

Properties:

K (x, x) = 1

(a point is maximally similar to itself)

K (x_{1}, x_{2}) \to 0

∥ x_{1} - x_{2} ∥ \to \infty

γ > 0

controls the kernel width — larger

γ

→ tighter, more local kernel

Your task:

Implement rbf_kernel(x1, x2, gamma) that returns the RBF kernel value between two vectors.

Example Tests

Identical points: K = 1.0

Input: {"x1":[1,2],"x2":[1,2],"gamma":0.5}

Expected: 1

Unit distance, gamma=1: K = e^-1 ≈ 0.3679

Input: {"x1":[0,0],"x2":[1,0],"gamma":1}

Expected: 0.36788

Larger gamma: faster decay

Input: {"x1":[0,0],"x2":[1,0],"gamma":2}

Expected: 0.13534

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