---
title: "19 Approximation de Nyström"
author: Pierre-Edouard Portier
date: mars 2022
output:
  beamer_presentation:
    incremental: false
---

```{r, include=FALSE}
source("01_intro_code.R", local = knitr::knit_global())
source("04_validation_croisee_code.R", local = knitr::knit_global())
source("18_kernel_ridge_regression_code.R", local = knitr::knit_global())
```

# Approximation de Nyström d'un noyau

>- $(\mathbf{x_i},y_i) \quad \text{avec} \quad i=1,\dots,n \quad ; \quad y_i \in \mathbb{R} \quad ; \quad \mathbf{x_i} \in \mathbb{R}^d$
>- Noyau $\mathbf{K}$
$$
\mathbf{K} =
\left[
\begin{array}{ccc}
k(\mathbf{x_1},\mathbf{x_1}) & \dots & k(\mathbf{x_1},\mathbf{x_n}) \\
\dots & \dots & \dots \\
k(\mathbf{x_n},\mathbf{x_1}) & \dots & k(\mathbf{x_n},\mathbf{x_n}) \\
\end{array}
\right]
$$
>- Approximation de rang $m$ de $\mathbf{K}$
$$
\mathbf{K} \approx \mathbf{U}\boldsymbol\Lambda\mathbf{U}^T \quad ; \quad \mathbf{U} \in \mathbb{R}^{n \times m} \quad ; \quad \boldsymbol\Lambda \in \mathbb{R}^{m \times m}
$$

# Approximation de Nyström d'un noyau

>- Sélection de $m$ observations qui sont les centres pour les calculs de similarités
$$
\mathbf{K} =
\left[
\begin{array}{cc}
\mathbf{K_{11}} & \mathbf{K_{21}^T}  \\
\mathbf{K_{21}} & \mathbf{K_{22}}  \\
\end{array}
\right]
\quad ; \quad
\mathbf{K_{11}} \in \mathbb{R}^{m \times m}
\quad ; \quad
\mathbf{K_{21}} \in \mathbb{R}^{(n-m) \times m}
$$

>- Objectif : ne pas évaluer $\mathbf{K_{22}}$

# Approximation de Nyström d'un noyau

>-
$$
\mathbf{K} \approx
\left[
\begin{array}{c}
\mathbf{U_1}  \\
\mathbf{U_2}  \\
\end{array}
\right]
\boldsymbol\Lambda
\left[
\begin{array}{c}
\mathbf{U_1}  \\
\mathbf{U_2}  \\
\end{array}
\right]^T
=
\left[
\begin{array}{cc}
\mathbf{U_1}\boldsymbol\Lambda\mathbf{U_1}^T & \mathbf{U_1}\boldsymbol\Lambda\mathbf{U_2}^T \\
\mathbf{U_2}\boldsymbol\Lambda\mathbf{U_1}^T & \mathbf{U_2}\boldsymbol\Lambda\mathbf{U_2}^T \\
\end{array}
\right]
$$

$$
\mathbf{U_1} \in \mathbb{R}^{m \times m}
\quad ; \quad
\mathbf{U_2} \in \mathbb{R}^{(n-m) \times m}
$$

>- $\mathbf{U_2} \approx \mathbf{K_{21}} \mathbf{U_1} \boldsymbol\Lambda^{-1}$

>- $\mathbf{K_{22}} \approx \left(\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2}\right) \left(\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2}\right)^T$

>- $\mathbf{K} \approx \boldsymbol\Phi \boldsymbol\Phi^T$
$$
\boldsymbol\Phi =
\left[
\begin{array}{c}
\mathbf{K_{11}}^{1/2}  \\
\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2}  \\
\end{array}
\right]
$$

# Calcul de l'approximation de Nyström

>-
$$
\mathbf{C} =
\left[
\begin{array}{c}
\mathbf{K_{11}}  \\
\mathbf{K_{21}}  \\
\end{array}
\right]
$$

>-
$$
\begin{aligned}
 & \mathbf{K} \approx \mathbf{C} \mathbf{K_{11}}^{-1} \mathbf{C}^T \\
= \{& \mathbf{K_{11}} = \mathbf{U}\boldsymbol\Sigma\mathbf{U}^T \} \\
 & \mathbf{K} \approx \mathbf{C} \mathbf{U}\boldsymbol\Sigma^{-1}\mathbf{U}^T \mathbf{C}^T \\
= \{& \mathbf{L} \triangleq \mathbf{C} \mathbf{U}\boldsymbol\Sigma^{-1/2} \} \\
 & \mathbf{K} \approx \mathbf{L}\mathbf{L}^T
\end{aligned}
$$

# Approximation de Nyström et régression ridge à noyau

>- $\mathbf{G} \triangleq \mathbf{K} + \lambda \mathbf{I_n}$
>- $\boldsymbol\alpha_\lambda = \mathbf{G}^{-1} \mathbf{y}$
>- $f(\mathbf{x}) = \sum_{i=1}^{n} \alpha_i k(\mathbf{x},\mathbf{x_i})$
>- $\mathbf{G}^{-1} \approx \lambda^{-1}\mathbf{I_n} - \lambda^{-1}\mathbf{L}\left(\lambda\mathbf{I_m}+\mathbf{L}^T\mathbf{L}\right)^{-1}\mathbf{L}^T$