Hamzi - Eurandom

Reproducing Kernel Hilbert Spaces
The important properties of reproducing kernels are
• K(x, y) is unique.
• ∀x, y ∈ X , K(x, y) = K(y, x) (symmetry).
• ∑m i,j=1 αiαjK(xi, xj) ≥ 0 for αi ∈ R and xi ∈ X (positive definitness).
• ⟨K(x, ·), K(y, ·)⟩H = K(x, y).
• A Mercer kernel is a continuous positive definite kernel.
• The fact that Mercer kernels are positive definite and symmetric reminds
us of similar properties of Gramians and covariance matrices. This is an
essential fact that we are going to use in the following.
• Examples of kernels: k(x, x ′ ) = ⟨x, x ′ ⟩ d , k(x, x ′ ) = exp ( − ||x−x′ || )
,
k(x, x ′ ) = tanh(κ⟨x, x ′ ⟩ + θ).
2σ 2
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 21 / 55