From 639e14824c0128ad11833b9a490e00156f173040 Mon Sep 17 00:00:00 2001 From: Pierre-Edouard Portier Date: Sun, 5 Sep 2021 16:45:04 +0200 Subject: [PATCH] =?UTF-8?q?Progr=C3=A8s=20sur=20la=20r=C3=A9daction=20du?= =?UTF-8?q?=20cours=20ML.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ML1_03_tikhonov.Rmd | 20 +++ ML1_03_tikhonov.pdf | Bin 206483 -> 216177 bytes ML1_11_projecteurs.Rmd | 60 +++++++- ML1_11_projecteurs.pdf | Bin 334018 -> 341650 bytes ML1_12_factorisation_qr.Rmd | 197 ++++++++++++++++++++++++++ ML1_12_factorisation_qr.pdf | Bin 0 -> 208428 bytes ML1_13_puissance_iteree_par_blocs.Rmd | 72 ++++++++++ ML1_13_puissance_iteree_par_blocs.pdf | Bin 0 -> 167886 bytes ML1_14_geometrie_ridge_svd.Rmd | 54 +++++++ ML1_14_geometrie_ridge_svd.pdf | Bin 0 -> 167621 bytes 10 files changed, 400 insertions(+), 3 deletions(-) create mode 100644 ML1_12_factorisation_qr.Rmd create mode 100644 ML1_12_factorisation_qr.pdf create mode 100644 ML1_13_puissance_iteree_par_blocs.Rmd create mode 100644 ML1_13_puissance_iteree_par_blocs.pdf create mode 100644 ML1_14_geometrie_ridge_svd.Rmd create mode 100644 ML1_14_geometrie_ridge_svd.pdf diff --git a/ML1_03_tikhonov.Rmd b/ML1_03_tikhonov.Rmd index fb9ba12..e0fa4ba 100644 --- a/ML1_03_tikhonov.Rmd +++ b/ML1_03_tikhonov.Rmd @@ -84,3 +84,23 @@ lcoef <- sapply(alphas, ridge, data, 7) matplot(alphas, t(lcoef), type=c("b"), pch=1, col=1:8, log="x") legend("topright", legend = 0:7, col=1:8, pch=1) ``` + +# Régularisation et complexité + +Essayons de mieux comprendre les raisons pour lesquelles la régularisation peut être efficace. Nous allons montrer qu'elle réduit la complexité d'un modèle prédictif. Dans ce contexte, qu'est-ce que la complexité ? C'est la sensibilité du modèle au bruit présent dans les données. Qu'est-ce que le bruit ? Il est possible de le définir après avoir fait l'hypothèse d'une famille de modèles qui auraient pu générer les données observées. + +Prenons l'exemple d'un modèle linéaire : $F(\mathbf{X}) = \mathbf{W}^T\mathbf{X} + b$. Posons ensuite $\mathbf{X^*} = \mathbf{X} + \mathbf{\epsilon}$ avec $\mathbf{\epsilon}$ un faible bruit ($\|\epsilon\|$ est petit). $\mathbf{X}$ et $\mathbf{X^*}$ étant proches, une hypothèse de régularité demande que $F(\mathbf{X})$ et $F(\mathbf{X^*})$ le soient aussi. La proximité de $F(\mathbf{X})$ et $F(\mathbf{X^*})$ peut se mesurer avec $|F(\mathbf{X}) - F(\mathbf{X^*})|$. + +\begin{align*} + & |F(\mathbf{X}) - F(\mathbf{X^*})| \\ += \{& F(\mathbf{X}) = \mathbf{W}^T\mathbf{X} + b \} \\ + & |\mathbf{W}^T\mathbf{X} - \mathbf{W}^T\mathbf{X^*}| \\ += \phantom{\{}& \\ + & |\mathbf{W}^T(\mathbf{X}-\mathbf{X^*})| \\ += \{& \mathbf{X^*} = \mathbf{X} + \mathbf{\epsilon} \} \\ + & |\mathbf{W}^T \mathbf{\epsilon}| \\ +\leq \{& \text{Inégalité de Cauchy-Schwarz} \} \\ + & \|\mathbf{W}\|_2 \|\mathbf{\epsilon}\|_2 \\ +\end{align*} + +Donc, en pénalisant les grandes valeurs de $\|\mathbf{W}\|_2$, on réduit la sensibilité du modèle à de petites perturbations dans les données observées. 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zg?3nk4(Nm~=!PEXg+Azq0Szwm_KZOd-fyjj)Wu&`!|JXstVYz`d{~XDyFa!XQ}>jy z8dsl-tR~c7KUhtwe~YcA)PMe;POJY;S &: \text{l'espace (la ligne) couvert par la première colonne de $\mathbf{A}$} \\ +<\mathbf{a_1}, \mathbf{a_2}> &: \text{l'espace (le plan) couvert par les deux premières colonne de $\mathbf{A}$} \\ +\text{Etc.} &: \text{Etc.} \\ +\end{align*} + +Nous avons $<\mathbf{a_1}> \subseteq <\mathbf{a_1}, \mathbf{a_2}> \subseteq \dots \subseteq <\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_N}>$. + +Nous cherchons des vecteurs orthogonaux $\mathbf{q_i}$ tels que : +\[ +<\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_j}> = <\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_j}> \quad \text{pour } j=1,2,\dots,N +\] + +Ce qui peut s'écrire sous une forme matricielle : +\[ +(\mathbf{a_1} \; \mathbf{a_2} \; \dots \; \mathbf{a_N}) = +(\mathbf{q_1} \; \mathbf{q_2} \; \dots \; \mathbf{q_N}) +\left( \begin{array}{cccc} +r_{11} & r_{12} & \dots & r_{1N} \\ +0 & r_{22} & \dots & r_{2N} \\ +\dots & \dots & \dots & \dots \\ +0 & 0 & \dots & r_{NN} +\end{array} \right) +\] + +Sans perte de généralité, considérons la situation où $M>N$. Nous appelons alors cette expression, $\mathbf{A}=\mathbf{\hat{Q}}\mathbf{\hat{R}}$ (avec $\mathbf{\hat{Q}} \in \mathbb{R}^{M \times N}$ et $\mathbf{\hat{R}} \in \mathbb{R}^{N \times N}$), la forme réduite de la factorisation QR. + +Nous pouvons aussi considérer la forme dite complète de la factorisation QR, avec $\mathbf{Q} \in \mathbb{R}^{M \times M}$ et $\mathbf{R} \in \mathbb{R}^{M \times N}$. Dans ce cas, les $M-N$ dernières lignes de $\mathbf{R}$ sont nulles et il suffit de compléter $\mathbf{\hat{Q}}$ en lui ajoutant $M-N$ colonnes orthogonales. + +# Orthogonalisation de Gram-Schmidt + +Nous présentons l'algorithme de Gram-Schmidt pour la décomposition QR. + +Le principe est de construire $\mathbf{q_j}$ orthogonal à $\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_{j-1}}$ en retirant à la colonne $\mathbf{a_j}$ ses composantes selon les directions de $\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_{j-1}}$ : + +\begin{align*} +\mathbf{v_j} &= \mathbf{a_j} - (\mathbf{q_1}^T\mathbf{a_j})\mathbf{q_1} - (\mathbf{q_2}^T\mathbf{a_j})\mathbf{q_2} - \dots - (\mathbf{q_{j-1}}^T\mathbf{a_j})\mathbf{q_{j-1}} \\ +\mathbf{q_j} &= \mathbf{v_j} / \|\mathbf{v_j}\| \\ +\end{align*} + +Nous obtenons ainsi la matrice $\mathbf{\hat{Q}}$. Pour $\mathbf{\hat{R}}$, nous avons, par construction des $\mathbf{q_j}$ : + +\begin{align*} +r_{ij} &= \mathbf{q_i}^T \mathbf{a_j} &\phantom{,} i \neq j \\ +|r_{jj}| &= \| \mathbf{a_j} - \sum_{i=1}^{j-1} r_{ij} \mathbf{q_i} \| &\phantom{,} \text{par convention, nous choisissons $r_{jj}>0$} \\ +\end{align*} + +Nous avons ainsi dérivé l'algorithme classique de Gram-Schmidt : + +\begin{algorithm}[H] +\DontPrintSemicolon +\NoCaptionOfAlgo +\SetAlgoLined +\SetKwInput{Res}{Résultat} +\SetKwInOut{Input}{Entrée}\SetKwInOut{Output}{Sortie} +\SetKwFor{Pour}{pour}{faire}{finpour} +\SetKw{KwTo}{à} +\Res{Décomposition QR de la matrice A} +\Input{$\mathbf{A} \in \mathbb{R}^{M \times N}$, avec $M \geq N$} +\Output{$\mathbf{Q} \in \mathbb{R}^{M \times N}$ une matrice orthonormale \\ +$\mathbf{R} \in \mathbb{R}^{N \times N}$ une matrice triangulaire supérieure \\ +telles que $\mathbf{A} = \mathbf{Q} \mathbf{R}$} +\BlankLine +\Pour{$j \gets 1$ \KwTo $N$}{ + $\mathbf{v_j} \gets \mathbf{a_j}$ \; + \Pour{$i \gets 1$ \KwTo $j-1$}{ + $r_{ij} \gets \mathbf{q_i}^T\mathbf{a_j}$ \; + $\mathbf{v_j} \gets \mathbf{v_j} - r_{ij}\mathbf{q_i}$ \; + } + $r_{jj} \gets \| \mathbf{v_j} \|$ \; + $\mathbf{q_j} \gets \mathbf{v_j} / r_{jj}$ \; +} +\caption{Gram-Schmidt classique (GS)} +\end{algorithm} + +# Algorithme de Gram-Schmidt modifié + +## Dérivation de l'algorithme de Gram-Schmidt modifié + +Nous présentons une autre manière de voir ce résultat qui nous mènera à un algorithme équivalent mais numériquement plus stable. + +Nous pouvons considérer que $\mathbf{q_j}$ est obtenu en projetant $\mathbf{a_j}$ sur l'espace orthogonal à $<\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_{j-1}}>$, cet espace étant par construction égal à $<\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_{j-1}}>$ : +\[ +\mathbf{q_j} = \frac{\mathbf{P_j}\mathbf{a_j}}{\|\mathbf{P_j}\mathbf{a_j}\|} \quad ; \quad +\mathbf{P_j} = \mathbf{I} - \mathbf{Q_{j-1}}\mathbf{Q_{j-1}}^T \; \text{avec } +\mathbf{Q_{j-1}} = (\mathbf{q_1} \; \mathbf{q_2} \; \dots \mathbf{q_{j-1}}) +\] + +De façon équivalente, $\mathbf{P_j}$ peut s'exprimer comme le produit de projecteurs sur les espaces orthogonaux à $\mathbf{q_1}$, puis $\mathbf{q_2}$, etc. jusqu'à $\mathbf{q_{j-1}}$ : +\[ +\mathbf{P_j} = \mathbf{P_{\bot \mathbf{q_{j-1}}}} \mathbf{P_{\bot \mathbf{q_{j-2}}}} \dots + \mathbf{P_{\bot \mathbf{q_{2}}}} \mathbf{P_{\bot \mathbf{q_{1}}}} \quad + \text{avec } \mathbf{P_{\bot \mathbf{q}}} = \mathbf{I} - \mathbf{q}\mathbf{q}^T +\] + +Dans la version classique de Gram-Schmidt, $\mathbf{v_j}=\mathbf{P_j}\mathbf{a_j}$. + +Dans la version modifiée de Gram-Schmidt : +\[ +\mathbf{v_j}=\mathbf{P_{\bot \mathbf{q_{j-1}}}} \mathbf{P_{\bot \mathbf{q_{j-2}}}} \dots + \mathbf{P_{\bot \mathbf{q_{2}}}} \mathbf{P_{\bot \mathbf{q_{1}}}} + \mathbf{a_j} +\] + +Ainsi, nous observons que dès que $\mathbf{q_1}$ est connu ($\mathbf{q_1} = \mathbf{v_1} / \|\mathbf{v_1}\|$, $\mathbf{v_1} = \mathbf{a_1}$), nous pouvons appliquer $\mathbf{P_{\bot \mathbf{q_{1}}}}$ à $\mathbf{a_2}, \mathbf{a_3}, \dots, \mathbf{a_N}$. Puis, quand $\mathbf{q_2}$ est connu, nous pouvons appliquer $\mathbf{P_{\bot \mathbf{q_{2}}}}$ à $\mathbf{P_{\bot \mathbf{q_{1}}}}\mathbf{a_3}, \mathbf{P_{\bot \mathbf{q_{1}}}}\mathbf{a_4}, \dots, \mathbf{P_{\bot \mathbf{q_{1}}}}\mathbf{a_N}$. Etc. Nous en déduisons l'algorithme de Gram-Schmidt modifié. + +\begin{algorithm}[H] +\DontPrintSemicolon +\NoCaptionOfAlgo +\SetAlgoLined +\SetKwInput{Res}{Résultat} +\SetKwInOut{Input}{Entrée}\SetKwInOut{Output}{Sortie} +\SetKwFor{Pour}{pour}{faire}{finpour} +\SetKw{KwTo}{à} +\Res{Décomposition QR de la matrice A} +\Input{$\mathbf{A} \in \mathbb{R}^{M \times N}$, avec $M \geq N$} +\Output{$\mathbf{Q} \in \mathbb{R}^{M \times N}$ une matrice orthonormale \\ +$\mathbf{R} \in \mathbb{R}^{N \times N}$ une matrice triangulaire supérieure \\ +telles que $\mathbf{A} = \mathbf{Q} \mathbf{R}$} +\BlankLine +\Pour{$i \gets 1$ \KwTo $N$}{ + $\mathbf{v_i} \gets \mathbf{a_i}$ \; +} +\Pour{$i \gets 1$ \KwTo $N$}{ + $r_{ii} \gets \|\mathbf{v_i}\|$ \; + $\mathbf{q_i} \gets \mathbf{v_i} / r_{ii}$ \; + \Pour{$j \gets i+1$ \KwTo $N$}{ + $r_{ij} \gets \mathbf{q_i}^T\mathbf{v_j}$ \; + $\mathbf{v_j} \gets \mathbf{v_j} - r_{ij}\mathbf{q_i}$ \; + } +} +\caption{Gram-Schmidt modifié (MGS)} +\end{algorithm} + +## Comparaison de la stabilité numérique des deux versions de l'algorithme de Gram-Schmidt + +Observons comment de petites erreurs de calculs se propagent dans le cas de l'algorithme de Gram-Schmidt (GS). Mettons qu'une petite erreur ait été commise sur le calcul de $\mathbf{q_2}$ qui s'en trouve de ce fait n'être pas parfaitement perpendiculaire à $\mathbf{q_1}$ : +\[ +\mathbf{q_1}^T\mathbf{q_2} = \epsilon > 0 +\] + +Quelle est l'effet de cette erreur sur la valeur de $\mathbf{v_3}$ ? + +\begin{align*} +\mathbf{v_3} &= \mathbf{a_3} - (\mathbf{q_1}^T\mathbf{a_3})\mathbf{q_1} - + (\mathbf{q_2}^T\mathbf{a_3})\mathbf{q_2} \\ +\mathbf{q_2}^T\mathbf{v_3} &= \mathbf{q_2}^T\mathbf{a_3} - (\mathbf{q_1}^T\mathbf{a_3})\epsilon - + (\mathbf{q_2}^T\mathbf{a_3}) = - (\mathbf{q_1}^T\mathbf{a_3})\epsilon \\ +\mathbf{q_1}^T\mathbf{v_3} &= \mathbf{q_1}^T\mathbf{a_3} - (\mathbf{q_1}^T\mathbf{a_3}) - + (\mathbf{q_2}^T\mathbf{a_3})\epsilon = - (\mathbf{q_2}^T\mathbf{a_3})\epsilon \\ +\end{align*} + +L'erreur commise sur $\mathbf{q_2}$ s'est transmise à $\mathbf{q_3}$ qui n'est plus parfaitement orthogonal à $\mathbf{q_1}$ et $\mathbf{q_2}$. + +Pour l'algorithme de Gram-Schmidt modifié (MGS), nous avons : + +\begin{align*} +\mathbf{v_3}^{(0)} &= \mathbf{a_3} \\ +\mathbf{v_3}^{(1)} &= \mathbf{v_3}^{(0)} - \mathbf{q_1}^T\mathbf{v_3}^{(0)}\mathbf{q_1} \\ +\mathbf{v_3} &= \mathbf{v_3}^{(1)} - \mathbf{q_2}^T\mathbf{v_3}^{(1)}\mathbf{q_2} \\ +\mathbf{q_2}^T\mathbf{v_3} &= \mathbf{q_2}^T\mathbf{v_3}^{(1)} - \mathbf{q_2}^T\mathbf{v_3}^{(1)} = 0 \\ +\end{align*} + +Il n'y a pas d'erreur pour l'orthogonalité entre $\mathbf{q_3}$ et $\mathbf{q_2}$. Etudions maintenant l'erreur commise sur l'orthogonalité entre $\mathbf{q_3}$ et $\mathbf{q_1}$. + +\begin{align*} +\mathbf{q_1}^T\mathbf{v_3} &= \mathbf{q_1}^T\mathbf{v_3}^{(1)} - \mathbf{q_2}^T\mathbf{v_3}^{(1)}\epsilon \\ +\mathbf{q_2}^T\mathbf{v_3}^{(1)} &= \mathbf{q_2}^T\mathbf{v_3}^{(0)} - \mathbf{q_1}^T\mathbf{v_3}^{(0)}\epsilon \\ +\mathbf{q_1}^T\mathbf{v_3}^{(1)} &= \mathbf{q_1}^T\mathbf{v_3}^{(0)} - \mathbf{q_1}^T\mathbf{v_3}^{(0)} = 0 \\ +\mathbf{q_1}^T\mathbf{v_3} &= -(\mathbf{q_2}^T\mathbf{v_3}^{(0)} - \mathbf{q_1}^T\mathbf{v_3}^{(0)}\epsilon) = + -\mathbf{q_2}^T\mathbf{v_3}^{(0)}\epsilon + \mathbf{q_1}^T\mathbf{v_3}^{(0)}\epsilon^2 +\end{align*} + +Ainsi, comme pour la version classique de Gram-Schmidt, une erreur de l'ordre de grandeur de $\epsilon$ est commise sur l'orthogonalité entre $\mathbf{q_1}$ et $\mathbf{q_3}$. Par contre, il n'y a pas de propagation de l'erreur pour l'orthogonalité entre $\mathbf{q_2}$ et $\mathbf{q_3}$. C'est pourquoi l'algorithme de Gram-Schmidt modifié est numériquement plus stable que l'algorithme de Gram-Schmidt classique. + +# Autres algorithmes + +Notons qu'une autre approche très souvent employée pour le calcul de la décomposition QR est la méthode dite de Householder. Elle assure le maintien d'une quasi parfaite orthogonalité des colonnes de $\mathbf{Q}$. Il existe également l'approche dite des rotations de Givens, plus facilement parallélisable et permettant la mise à jour de la décomposition suite à l'ajout d'une ligne à la matrice $\mathbf{X}$. + +Par contre, Gram-Schmidt permet de calculer la forme réduite de la décomposition QR, alors que les algorithmes de Givens et Householder calculent la forme complète. 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