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I want to prove that the zero extension of a kernel is still a kernel.
I.e. for a set of vectors in D:
$$k_0(x,x')= \begin{cases}k(x,x') \ if\ x\in D \ and \ x'\in D\\0\ otherwise\end{cases}$$ $k(x,x')$ is a valid kernel.
In my mind it would be enough to say it is a valid kernel because $k(x,x')$ is valid. The zero extension doesn't change anything, since for vectors $\notin D$, $k_0$ still remains positive semi definite (namely equal to zero).

Am I missing something?

Thanks!

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  • $\begingroup$ anyone that can help? $\endgroup$ – Dahlai Nov 24 '15 at 12:01
  • $\begingroup$ Just note that if you have a square matrix of the form $B = \begin{pmatrix} A & 0 \\ 0 & 0 \end{pmatrix}$ then $det (B - \lambda I) = det(A - \lambda I) \lambda^s$, where $s$ equals size of $B$ minus size of $A$. So no new eigenvalues would appear except $\lambda = 0$. $\endgroup$ – Alexey Zaytsev Dec 15 '15 at 15:48
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If $k(x,x')$ is a valid kernel, then $k_0(x, x') = f(x)k(x, x')f(x')$ is also a valid kernel for some real-valued function $f$. To see this, write

$$k_0(x, x') = f(x)\langle \phi(x), \phi(x') \rangle f(x') = \langle f(x)\phi(x), f(x') \phi(x') \rangle = \langle \psi(x), \psi(x') \rangle $$

where $\phi(\cdot)$ is the underlying feature map associated with $k$, and $\psi(x) = f(x)\phi(x)$ is the feature map associated with $k_0$.

Assume that the domain of $k$ is $\mathcal{X}$ and $D \subseteq \mathcal{X}$. Then, your $k_0$ is a valid kernel because

$$k_0(x, x') = I(x \in D) k(x, x') I(x' \in D)$$ where $I(a) \in \{0,1\}$ is the indicator function which is 1 if $a$ is true. Just set the above $f(x) := I(x \in D)$ to see that $k_0$ is valid.

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