27 lines
No EOL
793 B
Markdown
27 lines
No EOL
793 B
Markdown
# RKHS
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## Kernel definition
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Let `X` be a non empty set. Let $k: \mathbb{X} \times \mathbb{X} \to \mathbb{R}$ symetric and positive definite.
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$$
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\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n},
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\forall c \in \mathbb{R}^{n},
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\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0
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$$
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$ K_{i, j} = k(x_{i}, x_{j}) $
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### 1.2 Reproducing Kernel
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Let `H` be a Hilbert space of functions of real value functions $ f: \mathbb{X} \to \mathbb{R} $ endowed with the inner product $ \langle ., . \rangle_{H} $ k is a reproducing kernel if :
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- $ \forall x \in \mathbb{X}, k(., x) \in H $
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- $ \forall f \in H, \forall x \in \mathbb{X}, f(x) = \langle f, k(., x) \rangle_{H} $
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`H` is called the Reproducing Kernel Hilbert Space (RKHS) associated to `k`.
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**remark:** $ f = K(., x) then k $ |