Answer:
To find the kernel matrix using the Gaussian kernel assuming that o² = 5 and given x₁=(2,3), x₂=(3,4), and x₃=(2,4), we can use the following formula:
K(xᵢ, xⱼ) = exp(- ||xᵢ-xⱼ||² / 2o²)
where ||xᵢ-xⱼ|| is the Euclidean distance between points xᵢ and xⱼ. So, to find the kernel matrix , we first need to calculate the pairwise distances between the three points:
||x₁-x₂||² = (3-2)² + (4-3)² = 2 ||x₁-x₃||² = (2-2)² + (4-3)² = 1 ||x₂-x₃||² = (2-3)² + (4-4)² = 1
Then, we can plug these distances into the Gaussian kernel formula:
K(x₁, x₁) = exp(-0 / 10) = 1 K(x₁, x₂) = exp(-2 / 10) ≈ 0.67 K(x₁, x₃) = exp(-1 / 10) ≈ 0.82
K(x₂, x₁) = exp(-2 / 10) ≈ 0.67 K(x₂, x₂) = exp(-0 / 10) = 1 K(x₂, x₃) = exp(-1 / 10) ≈ 0.82
K(x₃, x₁) = exp(-1 / 10) ≈ 0.82 K(x₃, x₂) = exp(-1 / 10) ≈ 0.82 K(x₃, x₃) = exp(-0 / 10) = 1
Therefore, the kernel matrix is:
[ 1 0.67 0.82 ]
K = [ 0.67 1 0.82 ] [ 0.82 0.82 1 ]
Note that the kernel matrix is symmetric and positive semi-definite, which are the desired properties for a valid kernel matrix.
Step-by-step explanation: