131 lines
3.3 KiB
Plaintext
131 lines
3.3 KiB
Plaintext
---
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title: "19 Approximation de Nyström"
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author: Pierre-Edouard Portier
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date: mars 2022
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output:
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beamer_presentation:
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incremental: false
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---
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```{r, include=FALSE}
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source("01_intro_code.R", local = knitr::knit_global())
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source("04_validation_croisee_code.R", local = knitr::knit_global())
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source("18_kernel_ridge_regression_code.R", local = knitr::knit_global())
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```
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# Approximation de Nyström d'un noyau
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>- $(\mathbf{x_i},y_i) \quad \text{avec} \quad i=1,\dots,n \quad ; \quad y_i \in \mathbb{R} \quad ; \quad \mathbf{x_i} \in \mathbb{R}^d$
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>- Noyau $\mathbf{K}$
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$$
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\mathbf{K} =
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\left[
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\begin{array}{ccc}
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k(\mathbf{x_1},\mathbf{x_1}) & \dots & k(\mathbf{x_1},\mathbf{x_n}) \\
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\dots & \dots & \dots \\
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k(\mathbf{x_n},\mathbf{x_1}) & \dots & k(\mathbf{x_n},\mathbf{x_n}) \\
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\end{array}
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\right]
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$$
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>- Approximation de rang $m$ de $\mathbf{K}$
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$$
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\mathbf{K} \approx \mathbf{U}\boldsymbol\Lambda\mathbf{U}^T \quad ; \quad \mathbf{U} \in \mathbb{R}^{n \times m} \quad ; \quad \boldsymbol\Lambda \in \mathbb{R}^{m \times m}
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$$
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# Approximation de Nyström d'un noyau
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>- Sélection de $m$ observations qui sont les centres pour les calculs de similarités
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$$
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\mathbf{K} =
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\left[
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\begin{array}{cc}
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\mathbf{K_{11}} & \mathbf{K_{21}^T} \\
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\mathbf{K_{21}} & \mathbf{K_{22}} \\
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\end{array}
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\right]
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\quad ; \quad
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\mathbf{K_{11}} \in \mathbb{R}^{m \times m}
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\quad ; \quad
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\mathbf{K_{21}} \in \mathbb{R}^{(n-m) \times m}
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$$
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>- Objectif : ne pas évaluer $\mathbf{K_{22}}$
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# Approximation de Nyström d'un noyau
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>-
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$$
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\mathbf{K} \approx
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\left[
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\begin{array}{c}
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\mathbf{U_1} \\
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\mathbf{U_2} \\
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\end{array}
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\right]
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\boldsymbol\Lambda
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\left[
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\begin{array}{c}
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\mathbf{U_1} \\
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\mathbf{U_2} \\
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\end{array}
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\right]^T
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=
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\left[
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\begin{array}{cc}
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\mathbf{U_1}\boldsymbol\Lambda\mathbf{U_1}^T & \mathbf{U_1}\boldsymbol\Lambda\mathbf{U_2}^T \\
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\mathbf{U_2}\boldsymbol\Lambda\mathbf{U_1}^T & \mathbf{U_2}\boldsymbol\Lambda\mathbf{U_2}^T \\
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\end{array}
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\right]
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$$
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$$
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\mathbf{U_1} \in \mathbb{R}^{m \times m}
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\quad ; \quad
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\mathbf{U_2} \in \mathbb{R}^{(n-m) \times m}
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$$
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>- $\mathbf{U_2} \approx \mathbf{K_{21}} \mathbf{U_1} \boldsymbol\Lambda^{-1}$
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>- $\mathbf{K_{22}} \approx \left(\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2}\right) \left(\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2}\right)^T$
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>- $\mathbf{K} \approx \boldsymbol\Phi \boldsymbol\Phi^T$
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$$
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\boldsymbol\Phi =
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\left[
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\begin{array}{c}
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\mathbf{K_{11}}^{1/2} \\
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\mathbf{K_{21}} \mathbf{K_{11}}^{-1/2} \\
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\end{array}
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\right]
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$$
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# Calcul de l'approximation de Nyström
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>-
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$$
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\mathbf{C} =
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\left[
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\begin{array}{c}
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\mathbf{K_{11}} \\
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\mathbf{K_{21}} \\
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\end{array}
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\right]
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$$
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>-
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$$
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\begin{aligned}
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& \mathbf{K} \approx \mathbf{C} \mathbf{K_{11}}^{-1} \mathbf{C}^T \\
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= \{& \mathbf{K_{11}} = \mathbf{U}\boldsymbol\Sigma\mathbf{U}^T \} \\
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& \mathbf{K} \approx \mathbf{C} \mathbf{U}\boldsymbol\Sigma^{-1}\mathbf{U}^T \mathbf{C}^T \\
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= \{& \mathbf{L} \triangleq \mathbf{C} \mathbf{U}\boldsymbol\Sigma^{-1/2} \} \\
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& \mathbf{K} \approx \mathbf{L}\mathbf{L}^T
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\end{aligned}
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$$
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# Approximation de Nyström et régression ridge à noyau
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>- $\mathbf{G} \triangleq \mathbf{K} + \lambda \mathbf{I_n}$
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>- $\boldsymbol\alpha_\lambda = \mathbf{G}^{-1} \mathbf{y}$
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>- $f(\mathbf{x}) = \sum_{i=1}^{n} \alpha_i k(\mathbf{x},\mathbf{x_i})$
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>- $\mathbf{G}^{-1} \approx \lambda^{-1}\mathbf{I_n} - \lambda^{-1}\mathbf{L}\left(\lambda\mathbf{I_m}+\mathbf{L}^T\mathbf{L}\right)^{-1}\mathbf{L}^T$ |