www-ai.cs.tu-dortmund.de/LEHRE/FACHPROJEKT/SS14/Papers/kernelmethods.pdf
Kernel methods in machine learning
inequality: If k is a positive definite kernel, and x1, x2 ∈X , then
k(x1, x2) 2 ≤ k(x1, x1) · k(x2, x2).(9)
2.2.1. Construction of the reproducing kernel Hilbert space. We now de- fine a map from X into the [...] (15)
By virtue of these properties, k is called a reproducing kernel (Aronszajn [7]).
Due to (15) and (9), we have
|f(x)|2 = |〈k(·, x), f〉|2 ≤ k(x,x) · 〈f, f〉.(16)
By this inequality, 〈f, f〉= 0 implies f = [...] Gaussian kernel
k′(x,x′) = k(x,x′)f(x)f(x′) = e−‖x−x′‖2/(2σ2).(20)
KERNEL METHODS IN MACHINE LEARNING 9
2.2.3. Properties of positive definite functions. We now let X = R d and
consider positive definite …