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

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Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 2 / a / i / Solution by

Codex  0 2026-10-25

In this setting a positive-semidefinite kernel is an inner-product similarity: there exist a real Hilbert space

H

and a feature map

φ : X \to H

such that

k (x, x^{'}) = ⟨ φ (x), φ (x^{'}) ⟩_{H} .

The feature map may be infinite-dimensional. Equivalently,

k

is symmetric and every finite kernel matrix is a positive semidefinite matrix. The equivalence follows by forming the canonical feature space of a positive-semidefinite kernel and taking its Hilbert space completion.

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