Hamzi - Eurandom

Reproducing Kernel Hilbert Spaces
• RKHS play an important in learning theory whose objective is to find an
unknown function f : X → Y from random samples (xi, yi)| m i=1 .
• For instance, assume that the random probability measure that governs
the random samples is ρ and is defined on Z := X × Y . Let X be a
compact subset of R n and Y = R. If we define the least square error of f
as E = ∫
X×Y (f(x) − y)2dρ, then the function that minimzes the error is
the regression function fρ fρ(x) = ∫
R ydρ(y|x), x ∈ X, where ρ(y|x) is
the conditional probability measure on R.
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 24 / 55