From bbe2dc9fd1c0774ebd8e9aaa549033f4b9b33ee1 Mon Sep 17 00:00:00 2001 From: Alexis BAYLET Date: Fri, 27 Mar 2026 14:41:31 +0100 Subject: [PATCH] Refactor RKHS document for clarity and formatting improvements --- RKHS.md | 17 ++++++++++------- 1 file changed, 10 insertions(+), 7 deletions(-) diff --git a/RKHS.md b/RKHS.md index 34c39c0..6a9980b 100644 --- a/RKHS.md +++ b/RKHS.md @@ -5,12 +5,7 @@ Let `X` be a non empty set. Let $k: \mathbb{X} \times \mathbb{X} \to \mathbb{R}$ symetric and positive definite. -$$ -\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n}, -\forall c \in \mathbb{R}^{n}, - -\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0 -$$ +$\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n}, \forall c \in \mathbb{R}^{n}, \sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0$ $ K_{i, j} = k(x_{i}, x_{j}) $ @@ -24,4 +19,12 @@ Let `H` be a Hilbert space of functions of real value functions $ f: \mathbb{X} `H` is called the Reproducing Kernel Hilbert Space (RKHS) associated to `k`. -**remark:** $ f = K(., x) then k $ \ No newline at end of file +**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} $ +