Exercise 12.6: Eigenfunctions for Linear Kernels
The kernel integral operator for the linear kernel \(k\) is equal to \begin{align} T_k (f) (x) = \int \langle x, z \rangle f(z) \isd \P(z) = \bigl\langle x , \int z f(z) \isd \P(z) \bigr\rangle \, , \end{align} for any \(f \in L^2 (\P)\). Therefore we can find a basis for \(T_k\) by taking functions of the form \(\phi_j (x) = \langle v_j , x \rangle\) with \(v_j \in S^{d - 1}\) an orthonormal basis for \(\R^d\). The vectors \(v_j\) can then be obtained by eigendecomposition of the covariance matrix: \begin{align} \int z \phi_j(z) \isd \P(z) = \int z z^\top \isd \P(z) \, v_j \, . \end{align}