matterhacker

Given a kernel, want to construct a feature space (aka space obtained by map from data to feature space) . Feature spaces that can be constructed from a given kernel are not unique (they can even have different dimensions) but vice versa is not true… see below. Want to show that this constructed feature space is a dot product Hilbert space (aka pre-Hilbert space). Can construct a reproducing kernel map and show that it has the properties of a dot product space. Also can use mercer thm to construct a mercer map and show that it’s a dot product space. Although a mercer map is a rkm (is a mm a subspace of a rkm??). Other notes:

• A reproducing kernel hilbert space uniquely determines a kernel.
• Think of the map to feature space as a function at a particular data point . that relates it to the other data points x: Phi(x)(.) = k(.,x)