Thursday, September 29, 2005

reproducing kernel hilbert space

After a talk by Jean-Francois yesterday I decided to learn a little bit about RKHS: Reproducing Kernel Hilbert Space. So what's this new vector space that everybody is talking about?

The RKHS is easily grasped when one draws the analogy betwen role of the Dirac Delta in L2 and the reproducing kernel in a RKHS. The problem is that the dirac delta isn't in L2. Because of this property, the reproducing kernel of L2 isn't in L2. Due to the insanely small support size of the dirac delta, some non-smooth functions are in L2. In fact, we need two different notions of convergence for L2. We need convergence in the mean and pointwise convergence.

In an RKHS, convergence in the mean implies pointwise convergence. In an RKHS, the reproducing kernel usually has some support and therefore only smooth functions lie in this space. The reproducing kernel of an RKHS is actually in the RKHS! Great!

Did I get that right? I have to read Michael Jordan's notes again.