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RBF Kernel Matrix

Compute the Radial Basis Function (RBF/Gaussian) kernel matrix.

The RBF kernel is: $K(x, y) = \exp\left(-\frac{\|x - y\|^2}{2\sigma^2}\right)$

Given a set of vectors $X$ (n points) and parameter $\sigma$ , compute the $n \times n$ kernel matrix.

Return the matrix with values rounded to 4 decimal places. The diagonal should be all 1.0 (each point compared to itself).

Input:

X = [[0, 0], [1, 0], [0, 1]]
sigma = 1.0

Output:

[[1.0, 0.6065, 0.6065], [0.6065, 1.0, 0.3679], [0.6065, 0.3679, 1.0]]

Reasoning:

The RBF kernel matrix is computed by calculating $K(x, y) = \exp\left(-\frac{\|x - y\|^2}{2\sigma^2}\right)$ for each pair of points $x$ and $y$ in the input set $X$ .
For the given input $X = [[0, 0], [1, 0], [0, 1]]$ and $\sigma = 1.0$ , we calculate the pairwise distances and apply the RBF kernel formula to obtain the kernel matrix values.
The diagonal elements are all $1.0$ since $\|x - x\|^2 = 0$ , resulting in $K(x, x) = \exp(0) = 1.0$ for each point $x$ .
The off-diagonal elements are calculated using the RBF kernel formula, e.g., $K([0, 0], [1, 0]) = \exp\left(-\frac{\|(0-1)^2 + (0-0)^2\|}{2 \cdot 1.0^2}\right) = \exp\left(-\frac{1}{2}\right) \approx 0.6065$ .

Editor

Python 3.13.1

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