 Solution

ID: past-exam-of-the-advanced-topics-in-statistical-machine-learning-course-of-the-university-of-oxford/2025/2/a/iii/solution

φ_{1}, φ_{2}

Nonnegative sums: $k = α k_{1} + β k_{2}$ , for $α, β \geq 0$ . Its feature map is $(α φ_{1} (x), β φ_{2} (x))$ in the Hilbert space direct sum; expanding its inner product gives $k$ .
Pointwise products: $k (x, x^{'}) = k_{1} (x, x^{'}) k_{2} (x, x^{'})$ . In the Hilbert tensor product, use $φ (x) = φ_{1} (x) \otimes φ_{2} (x)$ : the inner product of two such elementary tensors is the product of their two inner products.
Input-dependent weighting: $k (x, x^{'}) = a (x) k_{1} (x, x^{'}) a (x^{'})$ for any real-valued function $a$ . Its feature map is $a (x) φ_{1} (x)$ , so signs of $a$ are allowed. Nonnegative scalar multiplication is the special case of constant $a$ ; the weighting factor $a (x) a (x^{'})$ is itself a rank-one positive-semidefinite kernel.

Another useful consequence is pullback: if

h : X \to Z

and

k_{0}

Z

, then

k_{0} (h (x), h (x^{'}))

φ_{0} \circ h

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