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Properties of Kernels [6* P]

For all the problems, let $ k$ be a positive definite kernel and $ X$ be the non-empty input space:
  1. Prove that if $ k(x,x) = 0$ for all $ x \in X$ , the kernel is identically zero, i.e. $ k(x,x') = 0  \forall x,x' \in X$ .
  2. Give an example of a kernel $ k$ which is positive definite, but not positive in the sense that $ k(x, x') \geq 0$ for all $ x,x' \in X$ . Give an example where the contrary is the case.
  3. Prove that the inhomogeneous polynomial $ k(x,x') = \left(\langle x,x' \rangle + c \right)^d$ with $ X \subset \mathbb{R}^N, d\in \mathbb{N}, c \geq 0$ is a positive definite kernel.
  4. Give an example of a kernel $ k$ with two valid feature maps $ \Phi_1, \Phi_2$ , mapping into spaces $ H_1, H_2$ of different dimensions.
  5. Show that a reproducing kernel $ k$ (i.e. $ \langle f, k(x,.) \rangle = f(x)$ for all $ f \in H$ , where $ H$ is a Hilbert space of functions $ f: X \rightarrow \mathbb{R}$ ), is symmetric.
  6. Given a kernel $ k$ , construct a corresponding normalized kernel $ \tilde{k}$ by normalizing the feature map $ \tilde{\Phi}$ such that for all $ x \in X, \Vert \tilde{\Phi}(x) \Vert = 1$ . Use this results to show that $ k(x,x') = cos(\angle(x,x'))$ is a positive definite kernel in a dot product space $ X$ .


next up previous
Next: About this document ... Up: MLA_Exercises_160106 Previous: Conditional Independence [2+2* P]
Pfeiffer Michael 2006-01-18