I am reading the Pattern Recognition and Machine Learning book by Bishops. I have some difficulties proving the (6.21) equation. Can someone help me? Image from Bishops book

N.B: If someone has a better title for this question, i'm opened to it as i couldn't find the good words.


Here are two proofs:

Gram matrix is psd: For any set of points $\{x_i\}_{i=1}^n$, call the Gram matrix for $k_a$ $A$ and the Gram matrix for $k_b$ $B$. Then the Gram matrix for $k$ is just $A + B$, and we have $$ \alpha^T (A + B) \alpha = \alpha^T A \alpha + \alpha^T B \alpha \ge 0 $$ since $\alpha^T A \alpha \ge 0$, $\alpha^T B \alpha \ge 0$. Thus $k$ is a psd kernel.

Feature maps: This is effectively taking the direct sum of two kernels. Let $\mathcal H_a$ be the RKHS for kernel $k_a$, and $\mathcal H_b$ for $k_b$, with feature maps $\varphi_a : \mathcal X_a \to \mathcal H_a$ and $\varphi_b : \mathcal X_b \to \mathcal H_b$. Define the RKHS $\mathcal H = \mathcal H_a \oplus \mathcal H_b$, where an arbitrary element of $\mathcal H$ is $f_a \oplus f_b$.

(If $\mathcal H_a = \mathbb R^{d_a}$ and $\mathcal H_b = \mathbb R^{d_b}$ are Euclidean spaces, then $\mathcal H = \mathbb R^{d_a + d_b}$ and $f_a \oplus f_b$ is just the concatenation of $f_a$ and $f_b$.)

The inner product is then $$\langle f_a \oplus f_b, f_a' \oplus f_b' \rangle_{\mathcal H} = \langle f_a, f_a' \rangle_{\mathcal H_a} + \langle f_b, f_b' \rangle_{\mathcal H_b}.$$ So we just need to define $\varphi(x) = \varphi_a(x_a) \oplus \varphi_b(x_b)$, in which case we get \begin{align} k( x, x' ) &= \langle \varphi(x), \varphi(x') \rangle_{\mathcal H} \\&= \langle \varphi_a(x_a) \oplus \varphi_b(x_b), \varphi_a(x_a') \oplus \varphi_b(x_b') \rangle_{\mathcal H} \\&= \langle \varphi_a(x_a), \varphi_a(x_a') \rangle_{\mathcal H_a} + \langle \varphi_b(x_b), \varphi_b(x_b') \rangle_{\mathcal H_b} \\&= k_a(x_a, x_a') + k_b(x_b, x_b') .\end{align} We can easily verify this a valid RKHS (sum of two continuous functionals is continuous), and so $k$ is again a psd kernel.

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