APM_4AI09/RKHS.md

# RKHS

## Kernel definition

Let `X` be a non empty set. Let $k: \mathbb{X} \times \mathbb{X} \to \mathbb{R}$ symetric and positive definite. 


$\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n}, \forall c \in \mathbb{R}^{n}, \sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0$

$ K_{i, j} = k(x_{i}, x_{j}) $

### 1.2 Reproducing Kernel

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 :

- $ \forall x \in \mathbb{X}, k(., x) \in H $
- $ \forall f \in H, \forall x \in \mathbb{X}, f(x) = \langle f, k(., x) \rangle_{H} $

`H` is called the Reproducing Kernel Hilbert Space (RKHS) associated to `k`.


**remark:** $ f = K(., x), k(x, x') = \langle k(., x'), k(., x) \rangle_{H} $

**Examples of kernels:**

### kernel PDS

- $k(x, x') = \exp(-\gamma \|x - x'\|^2) $ with $ x, x' \in \mathbb{X} $
- $k(x, x') = (1 + \langle x, x' \rangle)^{p} $ with $ x, x' \in \mathbb{X} $

## Moore Aronszajn Theorem (1943)

Let `k` be a PDS kernel over $ \mathbb{X} $.

There exists a Hilbert space `H` and a map $ \Phi: \mathbb{X} \to H $ such that $ \forall x, x' \in \mathbb{X}, k(x, x') = \langle \Phi(x), \Phi(x') \rangle_{H} $

Moreover, there is a unique Hilbert space such that k is a reproducing kernel of `H`.

Let's call `H` : `H_{k}`

- $ \forall x \in \mathbb{X}, k(., x) \in H $
- $ \forall f \in H, \forall x \in \mathbb{X}, f(x) = \langle f, k(., x) \rangle_{H} $