30 lines
1 KiB
Markdown
30 lines
1 KiB
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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$\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n}, \forall c \in \mathbb{R}^{n}, \sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0$
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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), k(x, x') = \langle k(., x'), k(., x) \rangle_{H} $
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## Examples of kernels
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### kernel PDS
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- $k(x, x') = \exp(-\gamma \|x - x'\|^2) $ with $ x, x' \in \mathbb{X} $
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- $k(x, x') = (1 + \langle x, x' \rangle)^{p} $ with $ x, x' \in \mathbb{X} $
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