From 3055621bb5fcaeda189aad10dfa34de17c6543ec Mon Sep 17 00:00:00 2001 From: Pierre-Edouard Portier Date: Mon, 30 Aug 2021 15:22:05 +0200 Subject: [PATCH] =?UTF-8?q?Nouvelle=20structuration=20en=20fichiers=20ind?= =?UTF-8?q?=C3=A9pendants=20avec=20code=20source=20dissoci=C3=A9.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ML1_01_intro.R | 50 ++++++++ ML1_01_intro.Rmd | 120 ++++++++++++++++++ ML1_01_intro.pdf | Bin 0 -> 204469 bytes ML1_02_moindres_carres.R | 8 ++ ML1_02_moindres_carres.Rmd | 88 +++++++++++++ ML1_02_moindres_carres.pdf | Bin 0 -> 207007 bytes ML1_03_tikhonov.R | 10 ++ ML1_03_tikhonov.Rmd | 88 +++++++++++++ ML1_03_tikhonov.pdf | Bin 0 -> 206483 bytes ML1_04_validation_croisee.R | 26 ++++ ML1_04_validation_croisee.Rmd | 52 ++++++++ ML1_04_validation_croisee.pdf | Bin 0 -> 154414 bytes ML1_05_presentation_svd.R | 20 +++ ML1_05_presentation_svd.Rmd | 143 ++++++++++++++++++++++ ML1_05_presentation_svd.pdf | Bin 0 -> 2591848 bytes ML1_06_matrice_definie_non_negative.Rmd | 73 +++++++++++ ML1_06_matrice_definie_non_negative.pdf | Bin 0 -> 131334 bytes ML1_07_multiplicateurs_de_lagrange.Rmd | 78 ++++++++++++ ML1_07_multiplicateurs_de_lagrange.pdf | Bin 0 -> 231014 bytes ML1_08_derivation_svd.Rmd | 83 +++++++++++++ ML1_08_derivation_svd.pdf | Bin 0 -> 131741 bytes ML1_09_puissance_iteree_valeur_propre.Rmd | 102 +++++++++++++++ ML1_09_puissance_iteree_valeur_propre.pdf | Bin 0 -> 189982 bytes ML1_10_puissance_iteree_svd.Rmd | 90 ++++++++++++++ ML1_10_puissance_iteree_svd.pdf | Bin 0 -> 166844 bytes ML1_11_projecteurs.Rmd | 84 +++++++++++++ ML1_11_projecteurs.pdf | Bin 0 -> 334018 bytes ML1_regression.Rmd | 26 ++-- ML1_regression.pdf | Bin 2160543 -> 2162163 bytes images/SVD_reduit.jpg | Bin 0 -> 111931 bytes images/projecteur1.jpg | Bin 0 -> 45995 bytes 31 files changed, 1132 insertions(+), 9 deletions(-) create mode 100644 ML1_01_intro.R create mode 100644 ML1_01_intro.Rmd create mode 100644 ML1_01_intro.pdf create mode 100644 ML1_02_moindres_carres.R create mode 100644 ML1_02_moindres_carres.Rmd create mode 100644 ML1_02_moindres_carres.pdf create mode 100644 ML1_03_tikhonov.R create mode 100644 ML1_03_tikhonov.Rmd create mode 100644 ML1_03_tikhonov.pdf create mode 100644 ML1_04_validation_croisee.R create mode 100644 ML1_04_validation_croisee.Rmd create mode 100644 ML1_04_validation_croisee.pdf create mode 100644 ML1_05_presentation_svd.R create mode 100644 ML1_05_presentation_svd.Rmd create mode 100644 ML1_05_presentation_svd.pdf create mode 100644 ML1_06_matrice_definie_non_negative.Rmd create mode 100644 ML1_06_matrice_definie_non_negative.pdf create mode 100644 ML1_07_multiplicateurs_de_lagrange.Rmd create mode 100644 ML1_07_multiplicateurs_de_lagrange.pdf create mode 100644 ML1_08_derivation_svd.Rmd create mode 100644 ML1_08_derivation_svd.pdf create mode 100644 ML1_09_puissance_iteree_valeur_propre.Rmd create mode 100644 ML1_09_puissance_iteree_valeur_propre.pdf create mode 100644 ML1_10_puissance_iteree_svd.Rmd create mode 100644 ML1_10_puissance_iteree_svd.pdf create mode 100644 ML1_11_projecteurs.Rmd create mode 100644 ML1_11_projecteurs.pdf create mode 100644 images/SVD_reduit.jpg create mode 100644 images/projecteur1.jpg diff --git a/ML1_01_intro.R b/ML1_01_intro.R new file mode 100644 index 0000000..87a0e28 --- /dev/null +++ b/ML1_01_intro.R @@ -0,0 +1,50 @@ +# ML 1 01 Intro + +# Exemple de fonction non-linéaire pour générer un jeu de données artificiel. +f <- function(x) {exp(x) * cos(2*pi*sin(pi*x))} + +# Génération d'un jeu de données synthétique qui est l'image par f d'un +# échantillon uniforme de l'intervalle [0,1] auquel nous ajoutons un bruit +# gaussien de moyenne nulle et d'écart type sd. +gendat <- function(N, sd) { + # N: nombre d'observations à générer + # sd: écart type d'un bruit gaussien de moyenne nulle + X = runif(N) + Y = f(X) + rnorm(N, mean=0, sd=sd) + dim(X) <- c(N,1) # en général chaque observation est décrite par plusieurs + # variables et X est une matrice avec autant de lignes que + # d'observations et autant de colonnes que de variables. Sur + # notre exemple, chaque observation n'est décrite que par une + # seule variable. + list(X = X, Y = Y) +} + +# Affichage simultanée du jeu de données bruité et de la courbe de la fonction +# utilisée pour le générer. +plt <- function(data, f, ...) { + xs = seq(0,1,length.out=100) + plot(xs, f(xs), type="l", ...) + points(data$X, data$Y) +} + +# Résolution du système linéaire correspondant à la matrice de Vandermonde. +# Autrement dit, découverte d'un polynôme qui passe par chaque point. +polyreg1 <- function(data) { + xs <- c(data$X) # on transforme la matrice X, de dimension Nx1 sur notre + # exemple, en un vecteur + vandermonde = outer(xs, 0:(length(xs)-1), "^") + solve(vandermonde, data$Y) +} + +# Evaluation d'un polynôme en un point. +polyeval <- function(coef,x) { + powers = 0:(length(coef)-1) + f = function(y) { sum(coef * y^powers) } + sapply(x,f) +} + +# Affichage de la courbe d'un polynôme défini par ses coefficients. +pltpoly <- function(coef) { + xs = seq(0,1,length.out=100) + lines(xs, polyeval(coef,xs), lty="dotted") +} \ No newline at end of file diff --git a/ML1_01_intro.Rmd b/ML1_01_intro.Rmd new file mode 100644 index 0000000..71a98fe --- /dev/null +++ b/ML1_01_intro.Rmd @@ -0,0 +1,120 @@ +--- +title: "ML 01 Introduction" +output: + pdf_document: + number_section: yes + html_notebook: + number_sections: yes +toc: false +classoption: fleqn +--- + +```{r, include=FALSE} +source("ML1_01_intro.R", local = knitr::knit_global()) +``` + +# Contexte + +A l’occasion des trois premières révolutions industrielles, des tâches, auparavant réservées au travail manuel de l’Homme, ont été automatisées. Il semble envisageable d’associer au tournant du 21ème siècle une quatrième révolution portée par l’automatisation de la capacité à prédire, essentielle pour le processus de prise de décision dans les secteurs de l’industrie, du commerce et des services. + +Cette transformation se fonde sur des évolutions scientifiques et techniques majeures. Elle est ainsi associée à une discipline, le machine learning ou apprentissage automatique de modèles prédictifs par extrapolation à partir de données générées par des processus physiques, numériques ou biologiques. Ces développements algorithmiques, en particulier la redécouverte des réseaux de neurones profonds, ont révélé sous un nouveau jour leur potentiel autour des années 2010 grâce, d’une part, à la création de jeux de données volumineux dans des domaines variés comme la reconnaissance de la parole, la vision par ordinateur, les données multimedia, le traitement de la langue naturelle, la robotique, les véhicules autonomes… et, grâce d’autre part, à une croissance rapide des capacités de calcul et de stockage aux coûts toujours plus abordables. + +Par ailleurs, cette automatisation des prédictions s’accompagne d’un renouveau des formes de jugement dans les processus de prise de décision avec un couplage de plus en plus fin entre d’un côté, des experts humains et de l’autre, des chaînes de traitement automatique des données qui aboutissent sur la mise en production d’algorithmes prédictifs. Pour la réussite de cette intégration, les compétences de l’ingénieur informaticien sont essentielles. Ce dernier, puisqu’il comprend en profondeur le fonctionnement et les limites des algorithmes qu’il déploie, est capable d’en mesurer les risques et les biais pour éclairer le jugement de ceux qui utiliseront ses réalisations logicielles pour prendre des décisions. + +Ainsi, ce cours introductif à l’apprentissage automatique a pour objectif d’offrir des connaissances fondamentales et des compétences pratiques qui aideront l'ingénieur à tenir ce rôle essentiel. + +# Objectifs + +La discipline de l’apprentissage automatique, ou machine learning, élabore des algorithmes et des méthodes pour découvrir des régularités dans des données multidimensionnelles afin, entre autres, d’automatiser la prédiction. Elle peut se subdiviser en trois catégories. D’abord, l’apprentissage non (ou semi) supervisé qui s’attache à découvrir des structures dans les données non étiquetées à travers des approches comme le clustering, la réduction dimensionnelle, les modèles génératifs… Ensuite, l’apprentissage par renforcement, dans le cadre duquel un agent interagit avec son environnement en adaptant son comportement pour maximiser une fonction de récompense. Enfin, l’apprentissage supervisé, qui fait l’objet de ce module, a quant à lui pour objectif d’apprendre à prédire l’association entre un objet décrit selon plusieurs dimensions et une étiquette. + +Par exemple, il peut s’agir d’associer aux quartiers d’une ville le prix médian d’un logement. Dans ce cas, un quartier peut être décrit par la proportion de zones résidentielles, le taux de criminalité, le nombre moyen de pièces par habitat, etc. Ici, nous faisons face à un problème dit de « régression » où la valeur à prédire, autrement l’étiquette associée à chaque observation, est continue. Lorsque la variable à prédire est discrète, il s’agit d’un problème dit de « classification », comme détecter un objet dans une image ou décider si une transaction bancaire risque d’être une fraude. Nous considérerons ces deux catégories de problèmes. + +Ainsi, ce cours a pour objectif d’introduire quelques concepts fondamentaux de l’apprentissage supervisé et de montrer leurs interconnexions variées dans le cadre de développements algorithmiques qui permettent d’analyser des jeux de données dans une visée avant tout prédictive. Ainsi, les propositions théoriques mèneront à l’écritures de programmes qui implémentent ou utilisent quelques modèles essentiels de l’apprentissage supervisé. + +Pour faciliter l’acquisition des connaissances, le cours est accompagné de notebooks manipulables, rédigés dans le langage de programmation R. Ces mises en pratique systématiques doivent permettre de faire le lien entre des concepts fondamentaux et leur application dans des projets d’analyse de données. + +# Ajustement de courbe + +$\mathbf{x}^{(1)} \dots \mathbf{x}^{(P)}$ sont des vecteurs de $\mathbb{R}^N$ associés aux valeurs, aussi appelées étiquettes, $y^{(1)} \dots y^{(P)}$ de $\mathbb{R}$. Nous cherchons une fonction $f(\mathbf{x}) : \mathbb{R}^N \rightarrow \mathbb{R}$ qui modélise la relation entre les observations $\mathbf{x}$ et les étiquettes $y$. + +La fonction $f$ peut avoir une forme paramétrique, comme par exemple : +\[ +f(\mathbf{x}) = a_0 + a_1x_1 + a_2x_2 + \dots + a_Nx_N +\] +Si $P=N+1$, les paramètres $a_0, a_1, \dots a_N$ sont solution d'un système linéaire : +\[ +\begin{cases} +y^{(1)} &= a_0 + a_1 x_1^{(1)} + a_2 x_2^{(1)} + \dots + a_N x_N^{(1)} \\ +y^{(2)} &= a_0 + a_1 x_1^{(2)} + a_2 x_2^{(2)} + \dots + a_N x_N^{(2)} \\ +\dots &= \dots \\ +y^{(P)} &= a_0 + a_1 x_1^{(P)} + a_2 x_2^{(P)} + \dots + a_N x_N^{(P)} \\ +\end{cases} +\] +Ce système s'écrit également sous forme matricielle : +\[ +\left( \begin{array}{cccc} +1 & x^{(1)}_1 & \dots & x^{(1)}_N \\ +1 & x^{(2)}_1 & \dots & x^{(2)}_N \\ +\dots & \dots & \dots & \dots \\ +1 & x^{(P)}_1 & \dots & x^{(P)}_N +\end{array} \right) +\left( \begin{array}{c} +a_0 \\ a_1 \\ \dots \\ a_N +\end{array} \right) += +\left( \begin{array}{c} +y^{(1)} \\ y^{(2)} \\ \dots \\ y^{(P)} +\end{array} \right) +\] +Chaque ligne $i$ de la matrice du terme de gauche de l'égalité ci-dessus est le vecteur ligne $\mathbf{x}^{(i)T}$ avec l'addition d'un premier terme constant qui correspond au paramètre $a_0$. En nommant cette matrice $\mathbf{X}^T$, le système linéaire ci-dessus s'écrit : +\[ +\mathbf{X}^T \mathbf{a} = \mathbf{y} +\] +Soit le cas particulier où $x$ est un scalaire et $f$ est un polynome de degré $N$ : +\[ +f(x) = a_0 + a_1x + a_2x^2 + \dots + a_Nx^N +\] +Avec $P=N+1$ observations et les étiquettes associées $\left( x^{(k)}, y^{(k)} \right)$, les coefficients de ce polynôme sont solution d'un système linéaire : + +\[ +\left( \begin{array}{ccccc} +1 & x^{(1)} & (x^{(1)})^2 & \dots & (x^{(1)})^N \\ +1 & x^{(2)} & (x^{(2)})^2 & \dots & (x^{(2)})^N \\ +\dots & \dots & \dots & \dots \\ +1 & x^{(P)} & (x^{(P)})^2 & \dots & (x^{(P)})^N +\end{array} \right) +\left( \begin{array}{c} +a_0 \\ a_1 \\ \dots \\ a_N +\end{array} \right) += +\left( \begin{array}{c} +y^{(1)} \\ y^{(2)} \\ \dots \\ y^{(P)} +\end{array} \right) +\] + +La matrice du terme de gauche de l'égalité ci-dessus est traditionnellement appelée "matrice de Vandermonde". + +# Interpolation polynomiale sur un jeu de données synthétique + +Soit un exemple de fonction non-linéaire, $f(x) = e^x \times cos(2 \pi \times sin(\pi x))$, utilisée pour générer un jeu de données synthétique. + +```{r} +set.seed(1123) +# Image par f d'un échantillon uniforme sur l'intervalle [0,1], avec ajout d'un +# bruit gaussien de moyenne nulle et d'écart type 0.2 +data = gendat(10,0.2) +plt(data,f) +``` + +Sur cet exemple, nous résolvons le système linéaire de Vandermonde pour découvrir un polynôme qui passe par chaque point du jeu de données. + +```{r} +# Résolution du système linéaire correspondant à la matrice de Vandermonde. +# coef contient les coefficients d'un polynôme qui passe par chaque point du jeu +# de données. +coef = polyreg1(data) +plt(data,f) +pltpoly(coef) +``` + +Il est improbable que ce polynôme, passant exactement par chaque observation, puisse offrir de bonnes capacités prédictives. Vérifier par exemple que, sur notre exemple synthétique, pour cinq points générés à partir de la fonction $f$ et avec l'ajout d'un bruit gaussien (par exemple d'écart type $0.2$), le polynôme découvert, de degré quatre, peut être très éloigné de la fonction génératrice. C'est un exemple du phénomène de sur-apprentissage. Pour limiter ce problème, nous cherchons à découvrir un polynôme de degré plus faible. 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Une approche possible est alors de chercher une approximation $\mathbf{Ax} \approx \mathbf{b}$ qui minimise la somme des carrés des erreurs : $\|\mathbf{Ax}-\mathbf{b}\|^2_2$. + +\begin{align*} + & \|\mathbf{A}\mathbf{x}-\mathbf{b}\|^2_2 \\ += \{ & \|\mathbf{x}\|_2 = \sqrt{\mathbf{x}\cdot\mathbf{x}} \} \\ + & \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \cdot \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ += \{ & \text{Par définition du produit scalaire euclidien} \} \\ + & \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right)^T \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ += \{ & \text{propriété de la transposition} \} \\ + & \left(\mathbf{x}^T\mathbf{A}^T - \mathbf{b}^T \right) \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ += \{ & \text{multiplication} \} \\ + & \mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x} - \mathbf{x}^T\mathbf{A}^T\mathbf{b} - \mathbf{b}^T\mathbf{A}\mathbf{x} + \mathbf{b}^T\mathbf{b} \\ += \{ & \mathbf{b}^T\mathbf{A}\mathbf{x} \text{ étant une valeur scalaire, } \mathbf{b}^T\mathbf{A}\mathbf{x} = \left(\mathbf{b}^T\mathbf{A}\mathbf{x}\right)^T = \mathbf{x}^T\mathbf{A}^T\mathbf{b} \} \\ + & \mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x} - 2\mathbf{x}^T\mathbf{A}^T\mathbf{b} + \mathbf{b}^T\mathbf{b} +\end{align*} + +Cette dernière expression quadratique en $\mathbf{x}$ correspond à une surface convexe. Donc son minimum peut être calculé en annulant sa dérivée (penser à une courbe $y = a+bx+cx^2$ dont l'unique extremum est atteint lorsque la pente est nulle). + +\begin{align*} + & \mathbf{0} = 2\mathbf{A}^T\mathbf{A}\mathbf{x} - 2\mathbf{A}^T\mathbf{b} \\ +=& \\ + & \mathbf{A}^T\mathbf{A}\mathbf{x} = \mathbf{A}^T\mathbf{b} +\end{align*} + +Ainsi, quand $M>N$, la solution approximée $\mathbf{x}$, telle que $\mathbf{Ax} \approx \mathbf{b}$ par minimisation de la somme des carrés des erreurs, est la solution du système linéaire suivant où $\mathbf{A}^T\mathbf{A}$ est appelée la matrice de Gram. +\[ +\mathbf{A}^T\mathbf{A} \mathbf{x} = \mathbf{A}^T\mathbf{b} +\] + +# Méthode des moindres carrés appliquée à la régression polynomiale + +Pour un polynôme de degré $N-1$, les fonctions de bases mentionnées ci-dessus sont : $f_1(x)=1$, $f_2(x)=x$, $f_3(x)=x^2$,..., $f_N(x)=x^{N-1}$. Elles permettent de définir la matrice des données $\mathbf{A}$ et la matrice de Gram $\mathbf{A}^T\mathbf{A}$. + +Nous reprenons l'exemple synthétique du précédent chapitre et nous résolvons le système linéaire correspondant à la matrice de Gram pour un polynôme de degré fixé. + +```{r} +set.seed(1123) +# Image par f d'un échantillon uniforme sur l'intervalle [0,1], avec ajout d'un +# bruit gaussien de moyenne nulle et d'écart type 0.2 +data = gendat(10,0.2) +coef = polyreg2(data,3) +plt(data,f) +pltpoly(coef) +``` + +Ce polynôme de degré trois modélise mieux la fonction génératrice inconnue que celui de degré quatre qui ne commettait aucune erreur sur les données observées. \ No newline at end of file diff --git a/ML1_02_moindres_carres.pdf b/ML1_02_moindres_carres.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3bd94c1f2d8aa2a1ac0d2e0034df227c7a982e7b GIT binary patch literal 207007 zcmd42V~{7$x-H!Hv^{NiPusTLJ#FjPwr$(CZQHgrZQI73|9;OtaU%8`aqh=kAF{Hl 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zYIc1IbBzlURHF{)W(xQs70`2QU+pqmF?$w>mJUmOCcy}h0IFY$b9mJQd{fQh{o55L ze02U!x^DNtOSX8OyGFnaK!Xq(uX|(mK-LzeQw@Pw=_B@REXw{_>M(Z;WIy*AtZa{L z$$flu=p;+`Tp@-xzx!rB0-SnslEC|JmyUl)K;S=B@hrf>KOn^aLIzS2e;G0@Kfi{d HChflf^Qe&@ literal 0 HcmV?d00001 diff --git a/ML1_04_validation_croisee.R b/ML1_04_validation_croisee.R new file mode 100644 index 0000000..c771aad --- /dev/null +++ b/ML1_04_validation_croisee.R @@ -0,0 +1,26 @@ +# Séparer le jeu de données en un jeu d'entraînement et un jeu de validation. +# INPUT : jeu de données initial et proportion des données conservées pour +# l'entraînement. +xvalpart <- function(data,p) { + n <- nrow(data$X) + nentr <- round(p*n) + entridx <- sample(1:n, nentr, replace=FALSE) + list(entr = list(X = data$X[entridx,,drop=FALSE], Y = data$Y[entridx]), + valid = list(X = data$X[-entridx,,drop=FALSE], Y = data$Y[-entridx])) +} + +# Découper le jeu de données initial en entraînement et validation. +# Pour chaque valeur proposée de l'hyperparamètre alpha d'une régression ridge, +# apprendre un modèle sur le jeu d'entraînement. +# Retourner les coefficients du meilleur modèle avec la moyenne de la valeur +# absolue des erreurs commises par ce modèle sur le jeu de validation. +# La fonction ridge est défini dans le code source de la partie 3. +xvalridge <- function(alphas, data, degre, p) { + tmp <- xvalpart(data,p) + lcoef <- sapply(alphas, ridge, tmp$entr, degre) + pred <- sapply(split(lcoef,col(lcoef)), polyeval, tmp$valid$X) + meanabserrs <- colMeans(abs(pred - tmp$valid$Y)) + minmeanabserr <- min(meanabserrs) + minidx <- which(meanabserrs == minmeanabserr) + list(meanabserr = minmeanabserr, coef = lcoef[,minidx], alpha = alphas[minidx]) +} \ No newline at end of file diff --git a/ML1_04_validation_croisee.Rmd b/ML1_04_validation_croisee.Rmd new file mode 100644 index 0000000..59526d7 --- /dev/null +++ b/ML1_04_validation_croisee.Rmd @@ -0,0 +1,52 @@ +--- +title: "ML 04 Validation croisée" +output: + pdf_document: + number_section: yes + html_notebook: + number_sections: yes +toc: false +classoption: fleqn +--- + +```{r, include=FALSE} +source("ML1_01_intro.R", local = knitr::knit_global()) +source("ML1_03_tikhonov.R", local = knitr::knit_global()) +source("ML1_04_validation_croisee.R", local = knitr::knit_global()) +``` + +# Principe de la validation croisée + +Comment choisir la valeur du coefficient de régularisation $\alpha$ pour une régression ridge ? Notons en passant que $\alpha$ est un exemple de ce que l'on appelle un hyperparamètre car sa valeur doit être fixée avant de pouvoir apprendre les paramètres du modèle (dans notre cas, les coefficients d'un modèle linéaire). + +Une possibilité est de diviser le jeu de données en deux parties, l'une utilisée pour apprendre le modèle prédictif, l'autre utilisée pour valider la qualité des prédictions sur des données qui n'ont pas été vues pendant la phase d'apprentissage. On parle de jeu d'entraînement et de jeu de validation ou de test. Cette méthode est appelée "validation croisée". + +Il s'agit de tester plusieurs valeurs de l'hyperparamètre $\alpha$ sur le jeu d'entraînement et de conserver celle qui donne les meilleurs résultats sur un jeu de données de validation qui n'a pas été utilisé pour l'entraînement. Nous obtenons ainsi un modèle qui montre souvent une meilleure capacité à prédire sur de nouvelles données. + +# Application de la validation croisée à la régularisation de Tikhonov + +Le code source accompagnant ce chapitre comprend une fonction `xvalpart` pour diviser le jeu de données en jeu d'entraînement et jeu de test. Ensuite, la fonction `xvalridge` apprend plusieurs modèles de régression ridge sur le jeu d'entraînement pour des valeurs différentes de l'hyperparamètre $\alpha$. Elle retourne les coefficients du meilleur modèle avec la moyenne de la valeur absolue des erreurs commises par ce modèle sur le jeu de validation. + +```{r} +set.seed(1123) +N <- 100 +deg1 <- 8 +data = gendat(N,0.2) +alphas <- c(1E-7, 1E-6, 1E-5, 1E-4, 1E-3, 1E-2, 1E-1, 1) +xvalres <- xvalridge(alphas, data, deg1, 0.8) +plt(data,f) +pltpoly(xvalres$coef) +``` + +Avec le code ci-dessus, nous générons un nouveau jeu de données composé de `r N` observations et nous calculons par validation croisée un polynôme de degré au plus égal à `r deg1` qui modélise au mieux ces données. La valeur de $\alpha$ retenue est : `r xvalres$alpha` et la moyenne des erreurs absolues est de `r xvalres$meanabserr`. + +Testons avec un polynôme de plus petit degré... + +```{r} +deg2 <- 5 +xvalres <- xvalridge(alphas, data, deg2, 0.8) +plt(data,f) +pltpoly(xvalres$coef) +``` + +Pour un polynôme de degré `r deg2`, la valeur de $\alpha$ retenue est : `r xvalres$alpha` et la moyenne des erreurs absolues est de `r xvalres$meanabserr`. Nous remarquons que pour un espace de fonctions plus riche (polynômes de degré `r deg1` VS polynômes de degré `r deg2`), une régularisation plus importante peut-être nécessaire, en particulier pour que le modèle ne soit pas trop sensible au bruit présent dans le jeu d'apprentissage. \ No newline at end of file diff --git a/ML1_04_validation_croisee.pdf b/ML1_04_validation_croisee.pdf new file mode 100644 index 0000000000000000000000000000000000000000..29bf39fb0eaad270de834f1c6c779bdc6c78e8f0 GIT binary patch literal 154414 zcmdSARa6~K_pgft2u>ilJHZz2?m>eS+$FfXy9aj&9zt-3g}ZBTcX!vlNb>&oJH~hM z?QzbICowHo6UpW9wh0w`Bn9V*aS|$w>NWowu*WgJ~!jNewgsc zDSPMLy1K}Mgb2S*$Y=z=rjEx&Tw=@+8o45x){2NEMnA=94($S2C71&G6Z`Yl)t8dM zu)yrIXl3oL+9De6X)USh%9H)I4B32*C)cMFYp($%GMOC22qGssnd7Ze#W*bTkENO= z`ATMw<~nso%`k#YPN(d08)3aY6E1ZFQ|l7nG6;&;JISqq9#A768}9%~x(tb9p#D(BT-Wv_U}V6J zc#t%hj+bPC0B2za()XIko*AYetY!c$H=KOHS!jQhs=zU`k5=dWh_^b5Xz+H73qs7t zjUB;IuuDxhNK~}rx_o5kQyRWf~;F?L^=(!o}{poA8(P=4TD79 ztYy05 zzr3v7n;<7cE0BdOgH4K`tIJi=D3@o|+;XCF znxa@;APG9IX1PGx9|(Vg>r_SNx+*PsG*j=5#N9P0g^cTqlK~3LjcmZF;%>WBD#(Af zgG8%a!g`bF}%K3iS%#Er%nV_$RhN0mJBo{)w10uwJH}I3QXYk^f%_Y{;c2)f5vk@AP;luXEcCs`OytzJ= zPd$Q7Hj--i9D0$5`}mk4LwsaPh4QX|%9uhPJD$P`PI8K3d1bzR+{E#t-YV48*T!1< z_RYlJ7d`8MCW{VZ{({5znQ5Hv@L*Nb!!&`ROA-&o=m|fdXa?Uf7cLJa6Vz+Aw;UTT zygeRh^K#heo64vgrV~}(Iu7K1uWU5L10zI*Z~wX8gRooj4V91Kk93>9pLY7P3U=gI z6tWcNg}sn6G34r!T&FsSMNFFWmiV-cioFl$W7ora--$7iSa?;&_4+3@`C`opKP2e} zNcWM~1|Rh0MX8L%?2_ZUGV6ZH+=AU3M@udX00V!jcrPFlN8UD}sjUks*4=NJiU=mj zEaWHekXr9tYDLjxM!nXVjcqAWfw1!F3<9EaCPE6}Z=aK9z?A$bWWTCE())-ZG|05W z^F_~Xh)vUQ+Y@H@^y^(Lv)RZdwO*muBtA5?=hDM?O}taN_RP^I0Q^-{uHg@A0aElw z62eh<$eB;hKP&yxNEe5LG9@3E6BWfSl^5UA_obs!HYoO+2;m}qDVKDqGT;q9*`tWH z=RIEpIK(8gko$k1?09=z{Edp9&9ild-LYRaC4V!KL*H*m0eCDuQieNYY}PjV@#CV5 z=3WW~+uJO6JFq;%SZXWL(u0>JC(F7CER>yvrrZ`BE}*83NXvH5_jEfVR*luq*%?(v zb4S-s3-cdHVu%^mp;qQSd26)laR&jzw-ZJllQmlv%z&;2rMNEw>a_2?fK`askkJQ$ zv`mvy)2?5?sY)ZlN#g{G67yX7*kN}tI9uAla=NeY%4BW^#liLqTx1}2E-oNs#P@D^ zi5q;E{e)RMzxFBm7@h7m41Pl~f>K!l%*+=dQv4*U$~x^MX@*(fgmM26B0X`kDt0ox zdy!tz$oZK)KX|)4Q$KtQnHNS8PvLG)bP(+Fr(4_Qa=xlnCT#~n%p+$~qDKzC_sO4r zjd7U=V7g(@4tT|^K1^18!cd}ur;VRg=P6Rbk#F132B3Rv4x`T+rW)=C3_tism93rz z9fAjmR+H||L4S{`SnV z{(ycvd>AHBRvG8Tz8U3-MIuuXu0jO;Ti%;|JB_=W>^YvO?x^bqwH8m50b%;~2n`S6 zs2!`P;o_aeCW$GNvWzSLOfZ6Mfa%fbcQvmXIh~RVMeI>VB-hnnT<);Mds1&10mLG$ z0Vr3fLYp)2u0JmBONX8!mzzu><(iX)U~yy@LtRNdgt**m5q0msqGv+ib}2n29xk+>NV>UrPt*Itk)Y^g6?IY^#vDzFm^X0>5sCA* zsOtq>RL_J1&dAE}Z!8BLz2*T-O#hh>aI&!c$p?u3dqPm6A{D*BhSKt&divG>&Chxd zMOrtraSDmA!l<{2M}iggrEs6^$joCN9}qa>RqcRBBzu9QDb@sd!<1sz0Wqq$iQA*N zNX(?^YJ&0T@r<%g!NW%fkd)q+X*X#rb!iJpIb=~>qzH4%XtF8E2^_3`X!+g_hA zo9wyl+dvC;l~IV?_sY5Wk1^b6Dcx*X+}T9bKD=A)7*@|@4hnys+zk%aJow4Q>Y0S_ zFgbUc{8HnIX4kewVlVj#35F58>Bo|cDMl63!47)7Hz{PRN;&S@*Hl`(7d37IrzlQ zA%&4o#D@`irS)b-j(KWecW7?2D!W9Rc#qFG&?}B!vV+O%SkQgS^GaDgEg2)K_92c6 z`bu9cGgfckL#akllk0(>NrrCMxne6<;6R!Z&@d*i5HG(W6bT+|*k3i2cY#kU>gu2l zyH~VN6>f{ob8U|tUN@}VoevUx4w&AwBe>oyDn>5z`gAaQlv9Well6g8A=nu$yDZS& zh`cXJgxS>4XQ!X{vVqWn>1uqI9A87lk7(jEH6GuaP4PTKQ{j$ zq~P-}f|pVVs5c*XBYyz{B9_Pqv+y}A%4Szr+IjX)MSnyc?WK;64?kpy@(Fpj2DNx) z?HF>p+lE#hJWaG_6&?Megfbe&E);6GOe4y?NF$Q?P_zj>x2Cmv%x4F2ilC4gj^C~Q zDN`zbhn`AqC7`skycn)-8++8S7M^?zrUEq(;t~4Ioy*jJ`ajFR z++WGn#)t@@Ca3?!$iM*(py;UY@cTbv)^?U4B>?^YeFY;N;FFT58a=31Fp|}?G$Nv+ z2gpb>>M*eAIO$oK8R|KhSzGBC*jby|8yV>sgRZg%7@L_m+8NQkRuwJ)M=L8(GebH9 zQ%5UvIwnwcVq<9h;qP5P>6sYGI$G+3s*ry=`eJr z+s2|U4&sUqpb7AL?$zkR))v-wiZ*%%zyAZUR}dE>VrKm3`hUF^c1}j-|N3ImYdob+ zETv>u9zDW%5g?sZc&C4;E24-9L4+>SF#1Hu`sHBY!n%F+c${q&FPWzcxTj{=ec-Ph zq(4yxENs~&&FC}_95|PG-<{Q!dAnxZJnWW@FE_W&C!fgcY|)=owLD*WH$NX=jq*Pq zoV+}EKUB?Ry_|2j5cZzUzdQ_Py~ybH-re)_KRDVhL;~IOKWc z-0G`y;Cl88Yz1s@-_FgK6J~xB>@(q<)&sAsZ@tb0?0nAtEgQ0eM*6pqQFh3U4mX$WrD z+}2MQLnn%58@#!#$%K!ds-uLNr=o5{FJrB2u-E&i8x)1t(nf-A3pzd{>uH!6QGDzM6?v)8*xVHLm|CIkJ<< z{kg4Vvu%^f_qimaAu@j1K812912B!Yb(Q?mK(OGB%4feCwj+1Ic$U+eoH%vzbBdvn95c#AMLlx^}vGEXI4^HpAC81(?v!cI@HVSd><@uR7;eJzVmA9_Pa_d~8FV z%-wO`Vn*H6s*FJ(rr5YWiP0C{H{;;O@!WOG=?t^Lk&FZ@k5|u452|p{ks&Af0fImRm^=n0s3CnD(dGtZB zyZszdlL?o#$sGhTWvmJ(aTQq4AMtxWGb#RJ7S38(eetq&e|XXE zVwEjitg$uDcYC?HX

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index 0000000..1426bd0 --- /dev/null +++ b/ML1_05_presentation_svd.Rmd @@ -0,0 +1,143 @@ +--- +title: "ML 05 Présentation de la décomposition en valeurs singulières (SVD)" +output: + pdf_document: + number_section: yes + html_notebook: + number_sections: yes +toc: false +classoption: fleqn +--- + +```{r, include=FALSE} +source("ML1_05_presentation_svd.R", local = knitr::knit_global()) +``` + +# Contexte + +Nous présentons la décomposition en valeurs singulières, une méthode générique de compression d'un tableau rectangulaire de données. Elle possède de nombreuses applications. En particulier, elle permet de calculer en un seul entraînement les régressions ridge pour toutes les valeurs possibles de l'hyperparamètre $\alpha$. Plus généralement, elle permet souvent d'adopter une perspective géométrique pour mieux comprendre les problèmes et algorithmes de l'apprentissage automatique à partir de données. + +# Recherche d'un sous-espace qui minimise les carrés des écarts à l'espace d'origine + +Soit un tableau de données $\mathbf{X}$ composé de $N$ observations en lignes, chacune décrite par $P$ variables en colonnes. $x_{ij}$ est la valeur de la variable $j$ pour l'observation $i$. Les vecteurs lignes forment $N$ points de l'espace $\mathbb{R}^P$. Les vecteurs colonnes forment $P$ points de l'espace $\mathbb{R}^N$. Nous cherchons à projeter le nuage des points lignes sur un sous-espace $\mathcal{H} \subset \mathbb{R}^P$, tout en minimisant les déformations. + +Pour commencer, nous considérons le meilleur sous-espace $\mathcal{H}$ à une dimension, c'est-à-dire une droite définie par son vecteur directeur unitaire $\mathbf{v}$ ("unitaire" signifie ici que sa norme euclidienne est égale à $1$, soit $\sqrt{\mathbf{v}^T\mathbf{v}}=1$, autrement dit, $\mathbf{v}^T\mathbf{v}=1$). Soit $M_i$ un des $N$ points de $\mathbb{R}^P$. A ce point correspond le vecteur $\mathbf{OM_i}$ aussi noté $\mathbf{x_i}$ car ses coordonnées se lisent sur la i-ème ligne de $\mathbf{X}$. Soit $H_i$ la projection de $M_i$ sur la droite $\mathcal{H}$. + +![svd1](images/svd1.jpeg) + +La longueur $OH_i$ est le produit scalaire de $\mathbf{x_i}$ et de $\mathbf{v}$. +\[ +OH_i = \mathbf{x_i}^T\mathbf{v} = \sum_{j=1}^{P} x_{ij} v_j +\] +Nous obtenons un vecteur des $N$ projections $OH_i$ par le produit de la matrice $\mathbf{X}$ et du vecteur $\mathbf{v}$ : $\mathbf{X}\mathbf{v}$. Choisissons pour $\mathcal{H}$ le sous-espace qui minimise la somme des carrés des écarts entre chaque point et sa projection : $\sum_i M_i H_i^2$. + +Le théorème de Pythagore donne $\sum_i M_i H_i^2 = \sum_i OM_i^2 - \sum_i OH_i^2$. + +Puisque $\sum_i OM_i^2$ est indépendant de $\mathbf{v}$, nous devons maximiser +\begin{equation} +\sum_i OH_i^2 = (\mathbf{X}\mathbf{v})^T(\mathbf{X}\mathbf{v}) = \mathbf{v}^T \mathbf{X}^T \mathbf{X} \mathbf{v} +\label{eq:svd-quadratic} +\end{equation} +...sous la contrainte $\mathbf{v}^T\mathbf{v}=1$. + +Nous notons $\mathbf{v_1}$ ce vecteur directeur du meilleur sous-espace de dimension 1 pour le nuage des $N$ observations de $\mathcal{R}^P$. Le meilleur sous-espace de dimension $2$ contient $\mathbf{v_1}$ et il faut le compléter avec un second vecteur unitaire $\mathbf{v_2}$, orthogonal à $\mathbf{v_1}$ et qui maximise $\mathbf{v_2}^T \mathbf{X}^T \mathbf{X} \mathbf{v_2}$. Etc. + +Nous montrerons plus loin que $\mathbf{v_1}$ est le vecteur propre de $\mathbf{X}^T \mathbf{X}$ associé à la plus grande valeur propre $\lambda_1$, cette dernière étant justement le maximum de la forme quadratique de l'équation (\ref{eq:svd-quadratic}). De même, $\mathbf{v_2}$ est le vecteur propre de $\mathbf{X}^T \mathbf{X}$ associé à la seconde plus grande valeur propre $\lambda_2$. Etc. + +Nous venons de voir que les vecteurs orthogonaux $\mathbf{v_1}, \mathbf{v_2},\dots, \mathbf{v_k}$ forment une base d'un sous-espace $k$-dimensionnel sur lequel peuvent être projetés les $N$ vecteurs lignes de $\mathbf{X}$ pour minimiser les carrés des écarts à l'espace d'origine qui était de dimension (au plus) $max(N,P)$. + +Nous construirons de même des vecteurs orthogonaux $\mathbf{u_1}, \mathbf{u_2},\dots, \mathbf{u_k}$ qui forment une base d'un sous-espace $k$-dimensionnel sur lequel peuvent être projetés les $P$ vecteurs colonnes de $\mathbf{X}$ pour minimiser les carrés des écarts à l'espace d'origine qui était de dimension (au plus) $max(N,P)$. + +Nous trouverons de façon similaire que $\mathbf{u_1}$ est le vecteur propre de $\mathbf{X} \mathbf{X}^T$ associé à la plus grande valeur propre $\mu_1$. De même, $\mathbf{u_2}$ est le vecteur propre de $\mathbf{X} \mathbf{X}^T$ associé à la seconde plus grande valeur propre $\mu_2$. Etc. + +Les vecteurs $\mathbf{u_1}, \mathbf{u_2}, \dots$ et $\mathbf{v_1}, \mathbf{v_2}, \dots$ sont les axes principaux de la matrice des données $\mathbf{X}$. + +Nous pouvons écrire : + +\begin{equation} +\begin{cases} +\mathbf{X}^T\mathbf{X}\mathbf{v_\alpha} = \lambda_\alpha \mathbf{v_\alpha} \\ +\mathbf{X}\mathbf{X}^T\mathbf{u_\alpha} = \mu_\alpha \mathbf{u_\alpha} +\end{cases} +\label{eq:svd-meme-valeurs-propres} +\end{equation} + +En prémultipliant la première des deux équations du système (\ref{eq:svd-meme-valeurs-propres}) par $\mathbf{X}$, nous obtenons : + +\[ +\left(\mathbf{X}\mathbf{X}^T\right)\mathbf{X}\mathbf{v_\alpha} = \lambda_\alpha \left(\mathbf{X}\mathbf{v_\alpha}\right) +\] + +Cette dernière équation nous indique qu'au vecteur propre $\mathbf{v_\alpha}$ de $\mathbf{X}^T\mathbf{X}$ correspond un vecteur propre $\left(\mathbf{X}\mathbf{v_\alpha}\right)$ de $\mathbf{X}\mathbf{X}^T$ de même valeur propre $\lambda_\alpha$. Comme $\mu_1$ est la plus grande valeur propre de $\mathbf{X}\mathbf{X}^T$, nous avons montré que $\lambda_1 \leq \mu_1$ + +De même, en prémultipliant la seconde des deux équations du système (\ref{eq:svd-meme-valeurs-propres}) par $\mathbf{X}^T$, nous montrerions que $\mu_1 \leq \lambda_1$. Donc, $\lambda_1 = \mu_1$ et en général, $\lambda_\alpha = \mu_\alpha$. + +Calculons la norme euclidienne, aussi appelée norme L2, de $\mathbf{X}\mathbf{v_\alpha}$. + +\begin{align*} + & \text{Norme euclidienne de } \mathbf{X}\mathbf{v_\alpha} \\ += \{ & \text{ Par définition de la norme euclidienne.} \} \\ + & \sqrt{\left(\mathbf{X}\mathbf{v_\alpha}\right)^T\left(\mathbf{X}\mathbf{v_\alpha}\right)} \\ += \{ &\text{Propriété de la transposition} \} \\ + & \sqrt{\mathbf{v_\alpha}^T\mathbf{X}^T\mathbf{X}\mathbf{v_\alpha}} \\ += \{ & \text{Voir équation (\ref{eq:svd-meme-valeurs-propres})} \} \\ +& \sqrt{\lambda_\alpha\mathbf{v_\alpha}^T\mathbf{v_\alpha}} \\ += \{ & \text{Par construction, } \mathbf{v_\alpha} \text{est de norme 1.} \} \\ +& \sqrt{\lambda_\alpha} \\ +\end{align*} + +Nous avons montré plus haut que $\left(\mathbf{X}\mathbf{v_\alpha}\right)$ est un vecteur propre de $\mathbf{X}\mathbf{X}^T$ de valeur propre associée $\lambda_\alpha$. Or, $\mathbf{u_\alpha}$ est le vecteur propre unitaire (i.e., de norme euclidienne égale à 1) de $\left(\mathbf{X}\mathbf{v_\alpha}\right)$ associé à la valeur propre $\lambda_\alpha$. C'est pourquoi, nous pouvons écrire : + +\[ +\begin{cases} +\mathbf{u_\alpha} = \frac{1}{\sqrt{\lambda_\alpha}} \mathbf{Xv_\alpha} \\ +\mathbf{v_\alpha} = \frac{1}{\sqrt{\lambda_\alpha}} \mathbf{X}^T\mathbf{u_\alpha} +\end{cases} +\] + +A partir de l'égalité $\mathbf{X}\mathbf{v_\alpha}=\mathbf{u_\alpha}\sqrt{\lambda_\alpha}$, nous post-multiplions par $\mathbf{v_\alpha}^T$ et nous sommons sur l'ensemble des $P$ axes principaux (en supposant, sans perte de généralité, que $N>P$) : + +\[ +\mathbf{X}\left(\sum_{\alpha=1}^{P}\mathbf{v_\alpha}\mathbf{v_\alpha}^T\right) = \sum_{\alpha=1}^{P}\sqrt{\lambda_\alpha}\mathbf{u_\alpha}\mathbf{v_\alpha}^T +\] + +Soit $\mathbf{V}$ la matrice formée des $P$ vecteurs $\mathbf{v_\alpha}$ en colonnes. Comme les vecteurs $\mathbf{v_\alpha}$ sont orthogonaux deux à deux et de norme $1$, $\mathbf{V}\mathbf{V}^T$ est la matrice identité $\mathbf{I_P}$. Donc, $\sum_{\alpha=1}^{P}\mathbf{v_\alpha}\mathbf{v_\alpha}^T=\mathbf{V}\mathbf{V}^T=\mathbf{I_P}$. Nous obtenons finalement : + +\begin{equation} +\mathbf{X} = \sum_{\alpha=1}^{P}\sqrt{\lambda_\alpha}\mathbf{u_\alpha}\mathbf{v_\alpha}^T +\label{eq:svd} +\end{equation} + +Nous avons montré que la matrice $\mathbf{X}$ peut s'écrire comme une somme de matrices de rang 1 qui sont les produits des vecteurs $\mathbf{u}$ (appelés *vecteurs singuliers à gauche*) et $\mathbf{v}$ (appelés *vecteurs singuliers à droite*) pondérés par les *valeurs singulières* $\sqrt{\lambda_\alpha}$. L'équation (\ref{eq:svd}) est celle de la *décomposition en valeurs singulières* de $\mathbf{X}$. + +Nous pouvons écrire ce résultat sous forme matricielle en introduisant les matrices $\mathbf{U}$, $\mathbf{V}$ et $\mathbf{D}$. Sans perte de généralité, nous exprimons les résultats pour $N>P$. La matrice $\mathbf{U}$, de dimension $N \times P$, contient en colonnes les vecteurs singuliers à gauche $\mathbf{u_1}, \mathbf{u_2}, \dots, \mathbf{u_P}$. La matrice $\mathbf{V}$, de dimension $P \times P$ contient en colonnes les vecteurs singuliers à droite $\mathbf{v_1}, \mathbf{v_2}, \dots, \mathbf{v_P}$. La matrice diagonale $\mathbf{D}$ contient sur sa diagonale les valeurs singulières $\sqrt{\lambda_1}, \sqrt{\lambda_2},\dots,\sqrt{\lambda_P}$. La décomposition en valeurs singulière de toute matrice $\mathbf{X}$ s'écrit alors : + +\[ +\mathbf{X} = \mathbf{U} \mathbf{D} \mathbf{V}^T +\] + +Notons que le rang de la matrice $\mathbf{X}$, c'est-à-dire le nombre de colonnes indépendantes (ou le nombre de lignes indépendantes, c'est équivalent), est égal au nombre de valeurs singulières non nulles. + +Par ailleurs, nous obtenons une reconstitution approchée de rang $k$ de $\mathbf{X}$ en ne conservant dans l'équation (\ref{eq:svd}) que les $k$ plus grandes valeurs singulières $\sqrt{\lambda_1},\sqrt{\lambda_2}, \dots, \sqrt{\lambda_k}$ et en annulant toutes les autres. + +## Illustrations du SVD avec la compression d'une image + +Considérons l'image d'un hortensia en fleurs. + +![hortensia](images/hortensia.jpg)\ + +Nous convertissons cette image, originellement au format JPEG, en un format en niveaux de gris, simple à manipuler, appelé PGM (*Portable Grey Map*). Dans ce format non compressé, l'image est représentée par une matrice dont chaque valeur, comprise entre 0 et 1, représente le niveau de gris d'un pixel. Nous opérons cette transformation grâce au programme *imagemagick* avec la commande `convert hortensia.jpg hortensia.pgm`. + +Ensuite, nous appliquons la décomposition en valeurs singulières à la matrice de l'image en niveaux de gris. Nous utilisons le résultat du SVD pour reconstituer une version compressée de l'image en ne conservant qu'un certain nombre des plus grandes valeurs singulières. + +```{r, warning=FALSE} +X <- read_grey_img("images/hortensia.pgm") + +par(mfrow=c(1,2), oma=c(1,1,0,0)+0.1, mar=c(0,0,1,1)+0.1); +print_grey_img(X, main="image originale d=256", asp=1); +print_grey_img(compress_SVD(X,16), main="d=16", asp=1); +print_grey_img(compress_SVD(X,32), main="d=32", asp=1); +print_grey_img(compress_SVD(X,64), main="d=64", asp=1); +print_grey_img(compress_SVD(X,128), main="d=128", asp=1); +print_grey_img(compress_SVD(X,256), main="d=256", asp=1); +``` diff --git a/ML1_05_presentation_svd.pdf b/ML1_05_presentation_svd.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ff8fbe98032bcb269e3ca3d2e7bba246844fe56d GIT binary patch literal 2591848 zcmeFZ2Ut_vwl*9DsbXk?)F?$%no3i|M5T)eHmVR6ktRf>MF<2zdfUJj1cabcrAe2T zNVg#!BnTv+QX?xWVM#*xmis&3z4zR^-T!~i+0TE^cmETfCt0MdtU1S+?|8>M#*C7c 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Pour que ce produit scalaire soit valide, $\mathbf{M}$ doit respecter la contrainte $\mathbf{x}^T\mathbf{M}\mathbf{y} \geq 0$ pour tout $\mathbf{x}$ et $\mathbf{y}$. + +Une fois donné un produit scalaire, les notions liées de norme et d'angle suivent~: $\|\mathbf{x}\|_{\mathbf{M}} = \sqrt{<\mathbf{x},\mathbf{x}>_{\mathbf{M}}}$ et $cos^2(\mathbf{x},\mathbf{y}) = <\mathbf{x},\mathbf{y}>_{\mathbf{M}}/\|\mathbf{x}\|_{\mathbf{M}}\|\mathbf{y}\|_{\mathbf{M}}$. + +La géométrie euclidienne "classique" (associée à la base canonique) correspond à l'emploi de la matrice identité pour métrique, $\mathbf{M}=\mathbf{I}$. + +Aussi, les valeurs propres d'une matrice PSD sont toutes positives~: + +\begin{align*} + & \mathbf{x}^T\mathbf{M}\mathbf{x} \geq 0 \\ +\Rightarrow \{ & \lambda \text{ est une valeur propre de } \mathbf{M} \} \\ + & \mathbf{x}^T\lambda\mathbf{x} \geq 0 \\ += \{ &\text{utilisation de la métrique identité} \} \\ + & \lambda \|x\|^2_{\mathbf{I}} \geq 0 \\ += \{ &\|x\|^2_{\mathbf{I}} \geq 0 \} \\ + & \lambda \geq 0 +\end{align*} + +Soit $\mathbf{V}$ la matrice des vecteurs propres de $\mathbf{M}$ et $\mathbf{L}$ la matrice diagonale de ses valeurs propres positives. + +\begin{align*} + & \mathbf{M}\mathbf{V} = \mathbf{V}\mathbf{L} \\ += \phantom{\{} & \\ + & \mathbf{M} = \mathbf{V}\mathbf{L}\mathbf{V}^{-1} \\ += \{ & \mathbf{S} = \sqrt{\mathbf{L}}\} \\ +& \mathbf{M} = \mathbf{V}\mathbf{S}\mathbf{S}\mathbf{V}^{-1} \\ += \{ & \quad \text{Les vecteurs propres distincts sont orthogonaux deux à deux : } \mathbf{V}^{-1} = \mathbf{V}^T. \\ +\phantom{=}\phantom{ }\phantom{\{} & \quad \mathbf{S}=\mathbf{S}^T. \text{On pose } \mathbf{T}=\mathbf{V}\mathbf{S}. \} \\ +& \mathbf{M} = \mathbf{T}\mathbf{T}^T \\ +\end{align*} + +Ainsi, toute matrice définie non négative $\mathbf{M}$ se décompose en $\mathbf{M}=\mathbf{T}\mathbf{T}^T$ où $\mathbf{T}$ est une transformation linéaire composée d'un changement d'échelle des axes du repère ($\mathbf{S}$) et d'une rotation ($\mathbf{V}$). + +Nous remarquons par exemple que l'équation d'un cercle selon la métrique $\mathbf{M}$ correspond à celle d'une ellipse selon la métrique identité $\mathbf{I}$ : + +\begin{align*} + & \|\mathbf{x}\|_{\mathbf{M}}^2 = c \\ += \{ &\text{C'est l'équation d'un cercle, c est une constante.}\} \\ + & < \mathbf{x}, \mathbf{x}>_{\mathbf{M}} = c \\ += \{ &\text{Par définition du produit scalaire.} \} \\ + & \mathbf{x}^T \mathbf{M} \mathbf{x} = c \\ += \{ &\mathbf{M} = \mathbf{T}^T\mathbf{T}\} \\ + & \mathbf{x}^T \mathbf{T}^T \mathbf{T} \mathbf{x} = c \\ += \{ &\text{Par définition du produit scalaire, on retrouve le cercle déformé en une ellipse par } \mathbf{T} \} \\ + & \|\mathbf{T}\mathbf{x}\|_{\mathbf{I}}^2 = c \\ +\end{align*} + +Ainsi, une matrice symétrique définie non négative projette le cercle unité sur une ellipse dont les axes orthonormaux sont les vecteurs propres et les longueurs des axes sont les valeurs propres. + +Enfin, l'angle formé entre $\mathbf{x}$ et $\mathbf{Mx}$ est inférieur ou égal à $\pi/2$ radians : + +\begin{align*} + & \mathbf{x}^T \mathbf{M} \mathbf{x} \geq 0 \\ += \{&\text{Géométrie du produit scalaire. On note } \theta \text{ l'angle entre les deux vecteurs.}\} \\ + & \|\mathbf{x}\|_\mathbf{I} \; \|\mathbf{Mx}\|_\mathbf{I} \; cos(\theta) \geq 0 \\ +\Rightarrow \phantom{\{}&\\ + & |\theta| \leq \pi/2 +\end{align*} + +Ainsi, le vecteur résultat $\mathbf{Mx}$ reste du même côté que $\mathbf{x}$ par rapport à l'hyperplan perpendiculaire à $\mathbf{x}$ qui sépare l'espace en deux. 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$\mathcal{R}^P$. Nous prouvons maintenant que $\mathbf{v_1}$ est le vecteur propre de $\mathbf{X}^T\mathbf{X}$ associé à la plus grande valeur propre $\lambda_1$. + +Le résultat est sans difficulté plus général. Nous le prouvons pour toute matrice symétrique $\mathbf{A}$ (non seulement $\mathbf{X}^T\mathbf{X}$) et toute métrique de $\mathcal{R}^P$ représentée par une matrice symétrique définie non négative $\mathbf{M}$ (non seulement la matrice identité $\mathbf{I}$). + +Nous rappelons que $\mathbf{v}^T\mathbf{A}\mathbf{v} = \Sigma a_{i,j}v_iv_j$. Ainsi, comme $\mathbf{A}$ et $\mathbf{M}$ sont symétriques (i.e., $a_{i,j}=a_{j,i}$ et $m_{i,j}=m_{j,i}$), les dérivées partielles des formes quadratiques ont les formes suivantes : +\[ +\frac{\partial \mathbf{v}^T\mathbf{A}\mathbf{v}}{\partial \mathbf{v}}=2\mathbf{A}\mathbf{v} \quad \text{et} \quad \frac{\partial \mathbf{v}^T\mathbf{M}\mathbf{v}}{\partial \mathbf{v}}=2\mathbf{M}\mathbf{v} +\] +Pour trouver le maximum de $\mathbf{v}^T\mathbf{A}\mathbf{v}$ en intégrant la contrainte unitaire sur $\mathbf{v}$ (i.e., $\mathbf{v}^T\mathbf{M}\mathbf{v}=1$), nous annulons les dérivées du langrangien $\mathcal{L}$ : + +\[ +\mathcal{L} = \mathbf{v}^T\mathbf{A}\mathbf{v} - \lambda (\mathbf{v}^T\mathbf{M}\mathbf{v} - 1) +\] + +\begin{align*} + & \frac{\partial \mathcal{L}}{\partial \mathbf{v}} = 0 \\ += \{& \text{voir calcul des dérivées ci-dessus} \} \\ + & 2\mathbf{A}\mathbf{v} - 2\lambda\mathbf{M}\mathbf{v} = 0 \\ += \phantom{\{} & \\ + & \mathbf{A}\mathbf{v} = \lambda\mathbf{M}\mathbf{v} \\ += \{& \mathbf{v}^T\mathbf{M}\mathbf{v}=1 \} \\ + & \lambda = \mathbf{v}^T\mathbf{A}\mathbf{v} +\end{align*} + +Nous découvrons que la valeur du multiplicateur de Lagrange $\lambda$ est le maximum recherché. Par ailleurs, nous avons : + +\begin{align*} + & \mathbf{A}\mathbf{v} = \lambda\mathbf{M}\mathbf{v} \\ += \{& \mathbf{M} \text{ est définie non négative et donc inversible} \} \\ + & \mathbf{M}^{-1}\mathbf{A}\mathbf{v} = \lambda\mathbf{v} +\end{align*} + +Donc $\mathbf{v}$ est le vecteur propre de $\mathbf{M}^{-1}\mathbf{A}$ associé à la plus grande valeur propre $\lambda$. Nous notons cette solution $\mathbf{v_1}$ et la valeur propre correspondante $\lambda_1$. + +Nous cherchons ensuite $\mathbf{v_2}$, unitaire ($\mathbf{v_2}^T\mathbf{M}\mathbf{v_2}=1$), orthogonal à $\mathbf{v_1}$ ($\mathbf{v_1}^T\mathbf{M}\mathbf{v_2}=0$) et qui maximise $\mathbf{v_2}^T\mathbf{A}\mathbf{v_2}$. Pour ce faire, nous annulons les dérivées du Langrangien ci-après. + +\[ +\mathcal{L} = \mathbf{v_2}^T\mathbf{A}\mathbf{v_2} - \lambda_2 (\mathbf{v_2}^T\mathbf{M}\mathbf{v_2} - 1) - \mu_2\mathbf{v_2}^T\mathbf{M}\mathbf{v_1} +\] + +\begin{align*} + & \frac{\partial \mathcal{L}}{\partial \mathbf{v_2}} = 0 \\ += \{& \text{Par définition de } \mathcal{L} \text{ et car A et M sont symétriques.}\} \\ + & 2\mathbf{A}\mathbf{v_2} - 2\lambda_2\mathbf{M}\mathbf{v_2} - \mu_2\mathbf{M}\mathbf{v_1}= 0 \\ +\Rightarrow \{& \quad \text{Multiplier à gauche chaque membre par } \mathbf{v_1}^T \\ +\phantom{=}\phantom{ }\phantom{\{} & \quad \mathbf{v_1}^T\mathbf{A}\mathbf{v_2}=0 \; ; \; \mathbf{v_1}^T\mathbf{M}\mathbf{v_2}=0 \; ; \; \mathbf{v_1}^T\mathbf{M}\mathbf{v_1}=1 \quad \} \\ + & \mu_2 = 0 +\end{align*} + +Nous avons utilisé ci-dessus la propriété $\mathbf{v_1}^T\mathbf{A}\mathbf{v_2}=0$ que nous prouvons ci-dessous. + +\begin{align*} + & \mathbf{v_1}^T\mathbf{A}\mathbf{v_2} \\ += \{& \mathbf{A} \text{ est symétrique.} \} \\ + & \mathbf{v_2}^T\mathbf{A}\mathbf{v_1} \\ += \{& \mathbf{A}\mathbf{v_1}=\lambda_1\mathbf{M}\mathbf{v_1} \text{ car } \mathbf{v_1} \text{ est vecteur propre de } \mathbf{M}^{-1}\mathbf{A} \} \\ + & \lambda_1 \mathbf{v_2}^T\mathbf{M}\mathbf{v_1} \\ += \{& \mathbf{v_1} \text{ et } \mathbf{v_2} \text{ sont M-orthogonaux.} \} \\ + 0 +\end{align*} + +Ainsi, nous trouvons : + +\begin{align*} + & \mathbf{A}\mathbf{v_2} = \lambda_2 \mathbf{M}\mathbf{v_2} \\ += \{& \mathbf{M} \text{ est définie non négative donc inversible.} \} \\ + & \mathbf{M}^{-1}\mathbf{A}\mathbf{v_2} = \lambda_2 \mathbf{v_2} +\end{align*} + +Donc, $\mathbf{v_2}$ est le vecteur propre de $\mathbf{M}^{-1}\mathbf{A}$ qui correspond à la seconde plus grande valeur propre $\lambda_2$. 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en valeurs propres d'une matrice carrée + +Dire que $\mathbf{u}$ est un vecteur propre de la matrice $\mathbf{A}$ associé à la valeur propre $\lambda$ signifie que $\mathbf{A}\mathbf{u}=\lambda\mathbf{u}$. Remarquons que tout multiple de $\mathbf{u}$ est également un vecteur propre associé à la même valeur propre $\lambda$. Considérons l'exemple de la matrice ci-dessous. +\[ +\mathbf{A}= +\left( \begin{array}{cccc} +1 & 2 & 0 \\ +2 & 1 & 0 \\ +0 & 0 & -1 +\end{array} \right) +\] +Notons $\mathbf{e_1}^T=(1 \; 0 \; 0)$, $\mathbf{e_2}^T=(0 \; 1 \; 0)$ et $\mathbf{e_3}^T=(0 \; 0 \; 1)$. Nous avons : + +\begin{align*} +\mathbf{A}\mathbf{e_3} &= -\mathbf{e_3} \\ +\mathbf{A}(\mathbf{e_1}+\mathbf{e_2}) &= 3(\mathbf{e_1}+\mathbf{e_2}) \\ +\mathbf{A}(\mathbf{e_1}-\mathbf{e_2}) &= -(\mathbf{e_1}-\mathbf{e_2}) \\ +\end{align*} + +Nous observons en particulier que la valeur propre $-1$ est associée à deux vecteurs propres linéairement indépendants : $\mathbf{e_3}$ et $(\mathbf{e_1}-\mathbf{e_2})$. + +# Plus grande valeur propre et méthode de la puissance itérée + +Posons $\mathbf{A}\in\mathbb{R}^{N\times N}$ et $\mathbf{u^{(0)}}\in\mathbb{R}^N$ avec $\|\mathbf{u^{(0)}}\|=1$. Supposons que $\mathbf{u^{(0)}}$ puisse se décomposer en une somme de vecteurs propres de $\mathbf{A}$. + +\begin{align*} +\mathbf{u^{(0)}} &= \mathbf{u_1}+\mathbf{u_2}+\dots +\mathbf{u_r} \\ +\text{Avec } \mathbf{A}\mathbf{u_i} &= \lambda_i \mathbf{u_i}\\ +\end{align*} + +Considérons la séquence $\mathbf{u^{(0)}}, \mathbf{A}\mathbf{u^{(0)}}, \mathbf{A}^2\mathbf{u^{(0)}}, \dots$. Nous avons : + +\begin{equation} +\mathbf{A}^m\mathbf{u^{(0)}} = \lambda_1^m \mathbf{u_1} + \lambda_2^m \mathbf{u_2} + \dots + \lambda_r^m \mathbf{u_r} +\label{eq:1} +\end{equation} + +Supposons par ailleurs que les valeurs propres soient ordonnées et que la première valeur propre soit strictement la plus grande en valeur absolue : $|\lambda_1|>|\lambda_2|\geq\dots\geq|\lambda_r|$. En divisant les deux membres de l'égalité (\ref{eq:1}) par $\lambda_1^m$, nous avons : +\[ +(\lambda_1^{-1}\mathbf{A})^m\mathbf{u^{(0)}}=\mathbf{u_1} + \sum_{i=2}^{r}\left(\frac{\lambda_i}{\lambda_1}\right)^m\mathbf{u_i} +\] +Or, $\left|\frac{\lambda_i}{\lambda_1}\right|<1$ pour $i=2,\dots,r$, donc nous pouvons conclure : +\[ +\lim\limits_{m \to \infty}(\lambda_1^{-1}\mathbf{A})^m\mathbf{u^{(0)}}=\mathbf{u_1} +\] + +En résumé, si $\mathbf{u^{(0)}}$ peut s'écrire comme une somme de vecteurs propres de $\mathbf{A}$ et que la valeur propre $\lambda_1$ associée au vecteur propre $\mathbf{u_1}$ est plus grande en valeur absolue que les autres valeurs propres, alors, à un facteur multiplicatif près, $\mathbf{A}^m\mathbf{u^{(0)}}$ converge vers le vecteur propre $\mathbf{u_1}$. + +Pour gérer ce facteur multiplicatif, nous considérons la séquence suivante : +\[ +\mathbf{w^{(k)}}=\mathbf{A}\mathbf{u^{(k-1)}} \quad ; \quad \mathbf{u^{(k)}}=\frac{\mathbf{w^{(k)}}}{\|\mathbf{w^{(k)}}\|} \quad ; \quad \lambda^{(k)}=\mathbf{u^{(k)}}^T(\mathbf{A}\mathbf{u^{(k)}}) +\] + +Par induction, nous obtenons : + +\begin{align*} + & \mathbf{u^{(k)}} \\ += \{& \gamma_k = \| \lambda_1^k\mathbf{u_1} + \sum_{j=2}^{r} \lambda_j^k\mathbf{u_j} \| \} \\ + & \frac{1}{\gamma_k} \left(\lambda_1^k\mathbf{u_1} + \sum_{j=2}^{r} \lambda_j^k\mathbf{u_j}\right) \\ += \phantom{\{}& \\ + & \frac{\lambda_1^k}{\gamma_k} \left(\mathbf{u_1} + \sum_{j=2}^{r} \left(\frac{\lambda_j}{\lambda_1}\right)^k \mathbf{u_j}\right) \\ +\end{align*} + +Par ailleurs, $\mathbf{u^{(k)}}$ étant à chaque itération normalisé, nous avons : + +\begin{align*} + & \|\mathbf{u^{(k)}}\| = 1 \\ += \phantom{\{}& \\ + & \frac{|\lambda_1^k|}{\gamma_k} = \frac{1}{\|\mathbf{u_1} + \sum_{j=2}^{r} \left(\frac{\lambda_j}{\lambda_1}\right)^k \mathbf{u_j}\|} +\end{align*} + +D'où : +\[ +\lim\limits_{k \to \infty} \frac{|\lambda_1^k|}{\gamma_k} = \frac{1}{\|\mathbf{u_1}\|} +\] + +Finalement : + +\[ +\lim\limits_{k \to \infty} \mathbf{u^{(k)}} = \frac{\mathbf{u_1}}{\|\mathbf{u_1}\|} +\] + +Et : + +\begin{align*} + & \lim\limits_{k \to \infty} \mathbf{u^{(k)}}^T(\mathbf{A}\mathbf{u^{(k)}}) \\ += \{& \mathbf{A}\mathbf{u_1}=\lambda_1\mathbf{u_1} \} \\ + & \lambda_1 \\ +\end{align*} \ No newline at end of file diff --git a/ML1_09_puissance_iteree_valeur_propre.pdf b/ML1_09_puissance_iteree_valeur_propre.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9108da84164188f8dc1ea009377b8f57bf77a977 GIT binary patch literal 189982 zcma%>Q;aYGvt-A%ZQHhO+qUN$+qP}nwr$(C_g~zny~%DrRVV#e>6}zIse*_YEh8N( z6zTl($T}1o0Rw@(krfmV4-~zOshzot1pyNa2Rp(4Dkyp}OB)wcCjxpg8$%aU5mRG( z6H_QYJ}74wCsRXPD38qa=Bx_w(#ROnIc;Zn8~$yGU3D#woO4o~m>*mOu^ z-cc0Jle|1T01`l`WR|N8CpsbtHhh>)_>AXtb zdID5!+dcL#Y`xk}EIP9l*03JG0PlmBgU}i{ZUfUk%b_B2a)8**%ANi2T0FQeT@9k? zs#f@)R=b}ouIkTaHQgs^e6-8A7QY@rOoNEBEY@lV4mM=`A({EO$4}0}YEaVF=;ESK zjH+&zF|)sQdu)xQ)|Urrl~mmu_@GtZsyCs`AwHgem0~r-8iMqb=h*Vy0>P7ADh 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singulière et de ses deux vecteurs singuliers associés. Nous considérons une matrice de données rectangulaire $\mathbf{A}\in\mathbb{R}^{N\times M}$. Nous notons $\mathbf{U}\in\mathbb{R}^{N\times N}$ et $\mathbf{V}\in\mathbb{R}^{M\times M}$ les deux matrices orthogonales qui contiennent les vecteurs singuliers en colonnes. Nous notons enfin $\mathbf{\Sigma} = diag(\sigma_1,\sigma_2,\dots,\sigma_r,0,\dots)\in\mathbb{R}^{N\times M}$ la matrice qui contient sur sa diagonale principale les valeurs singulières, avec $r$ le range de $\mathbf{A}$. Nous avons donc : + +\begin{align*} +\mathbf{A} &= \mathbf{U}\mathbf{\Sigma}\mathbf{V}^T & \text{$r$ est le rang de $\mathbf{A}$.} \\ +\mathbf{U} &= [\mathbf{u_1}, \mathbf{u_2},\dots,\mathbf{u_N}] & \text{avec $\mathbf{u_i}\in\mathbb{R}^N$, base orthonormale de $\mathbb{R}^N$} \\ +\mathbf{V} &= [\mathbf{v_1}, \mathbf{v_2},\dots,\mathbf{v_M}] & \text{avec $\mathbf{v_j}\in\mathbb{R}^M$, base orthonormale de $\mathbb{R}^M$} \\ +\mathbf{A} &= \sum_{k=1}^{r} \sigma_k \mathbf{u_k}\mathbf{v_k}^T & +\end{align*} + +Nous allons montrer que la séquence ci-dessous converge vers $\sigma_1$, $\mathbf{u}$ et $\mathbf{v}$ tels que $\mathbf{A}\mathbf{v}=\sigma_1\mathbf{u}$. + +\begin{align*} +\mathbf{w^{(k)}} &= \mathbf{A}\mathbf{v^{(k-1)}} & \alpha_k &= \|\mathbf{w^{(k)}}\|^{-1} & \mathbf{u^{(k)}} &= \alpha_k \mathbf{w^{(k)}} \\ +\mathbf{z^{(k)}} &= \mathbf{A}^T\mathbf{u^{(k)}} & \beta_k &= \|\mathbf{z^{(k)}}\|^{-1} & \mathbf{v^{(k)}} &= \beta_k \mathbf{z^{(k)}} \\ +err &= \|\mathbf{A}\mathbf{v^{(k)}} - \beta_k \mathbf{u^{(k)}}\| & \sigma_1 &= \beta_k & & +\end{align*} + +Nous décomposons $\mathbf{v^{(0)}}\in\mathbb{R}^M$ dans la base orthonormée formée par les vecteurs $\mathbf{v_j}$ : +\[ +\mathbf{v^{(0)}} = \sum_{j=1}^{M} y_j\mathbf{v_j} \quad \text{avec $y_j = \mathbf{v_j}^T\mathbf{v^{(0)}}$} +\] + +Partant de ce $\mathbf{v^{(0)}}$, nous calculons les deux premières itérations : + +\begin{align*} +\mathbf{w^{(1)}} &= \mathbf{A}\mathbf{v^{(0)}} = \sum_{j=1}^{r}\sigma_j y_j \mathbf{u_j} & \mathbf{u^{(1)}} &= \alpha_1 \sum_{j=1}^{r}\sigma_j y_j \mathbf{u_j} \\ +\mathbf{z^{(1)}} &= \mathbf{A}^T\mathbf{u^{(1)}} = \alpha_1 \sum_{j=1}^{r}\sigma_j^2 y_j \mathbf{v_j} & \mathbf{v^{(1)}} &= \alpha_1 \beta_1 \sum_{j=1}^{r}\sigma_j^2 y_j \mathbf{v_j} \\ +\mathbf{w^{(2)}} &= \mathbf{A}\mathbf{v^{(1)}} = \alpha_1 \beta_1 \sum_{j=1}^{r}\sigma_j^3 y_j \mathbf{u_j} & \mathbf{u^{(2)}} &= \alpha_2 \alpha_1 \beta_1 \sum_{j=1}^{r}\sigma_j^3 y_j \mathbf{u_j} \\ +\mathbf{z^{(2)}} &= \mathbf{A}^T\mathbf{u^{(2)}} = \alpha_2 \alpha_1 \beta_1 \sum_{j=1}^{r}\sigma_j^4 y_j \mathbf{v_j} & \mathbf{v^{(2)}} &= \alpha_2 \beta_2 \alpha_1 \beta_1 \sum_{j=1}^{r}\sigma_j^4 y_j \mathbf{v_j} \\ +\end{align*} + +Par induction, nous avons : + +\begin{align*} +\mathbf{v^{(k)}} &= \delta_{2k} \sum_{j=1}^{r} \sigma_j^{2k} y_j \mathbf{v_j} & \text{Avec } \delta_{2k} &= \alpha_k \beta_k \alpha_{k-1} \beta_{k-1} \dots \alpha_1 \beta_1 \\ +\mathbf{u^{(k)}} &= \delta_{2k-1} \sum_{j=1}^{r} \sigma_j^{2k-1} y_j \mathbf{u_j} & \text{Avec } \delta_{2k-1} &= \alpha_{k+1} \alpha_k \beta_k \alpha_{k-1} \beta_{k-1} \dots \alpha_1 \beta_1 \\ +\end{align*} + +Par ailleurs, nous avons : + +\begin{align*} + & 1 \\ += \{& \text{A chaque itération, la norme de $\mathbf{u^{(k)}}$ est maintenue égale à 1.} \} \\ + & \|\mathbf{u^{(k)}}\|^2 \\ += \{& \text{Par définition de $\mathbf{u^{(k)}}$ et car } \|\mathbf{u_j}\| = 1 \} \\ + & \delta_{2k-1}^2 \sum_{j=1}^{r} \sigma_j^{4k-2} y_j^2 \\ +\end{align*} + +De même : + +\begin{align*} + & 1 \\ += \phantom{\{}& \\ + & \|\mathbf{v^{(k)}}\|^2 \\ += \phantom{\{}& \\ + & \delta_{2k}^2 \sum_{j=1}^{r} \sigma_j^{4k} y_j^2 \\ +\end{align*} + +Donc : +\[ +\frac{\|\mathbf{u^{(k+1)}}\|^2}{\|\mathbf{v^{(k)}}\|^2} = 1 = +\sigma_1^2 \left(\frac{\delta_{2k+1}^2}{\delta_{2k}^2}\right) +\left(\frac{\sum_{j=1}^{n_1} y_j^2 \; + \; \sum_{j=n_1}^{r}\left(\frac{\sigma_j}{\sigma_1}\right)^{4k+2}y_j^2} +{\sum_{j=1}^{n_1} y_j^2 \; + \; \sum_{j=n_1}^{r}\left(\frac{\sigma_j}{\sigma_1}\right)^{4k}y_j^2}\right) +\] + +Ci-dessus, $n_1$ est la multiplicité de $\sigma_1$, c'est-à-dire le nombre de vecteurs singuliers indépendants associés à cette valeur singulière. En pratique, sur de "vraies" données, cette multiplicité sera presque toujours égale à $1$. + +Nous avons ainsi montré que : +\[ +\lim\limits_{k \to \infty} \frac{\delta_{2k+1}}{\delta_{2k}} = \sigma_1 +\] + +Et : +\begin{align*} + & \lim\limits_{k \to \infty} \|\mathbf{A}\mathbf{v^{(k)}} - \sigma_1 \mathbf{u^{(k)}}\| \\ += \{& \mathbf{A}\mathbf{v^{(k)}} = \frac{\delta_{2k+1}}{\delta_{2k}} \mathbf{u^{(k+1)}}\} \\ + & 0 +\end{align*} \ No newline at end of file diff --git a/ML1_10_puissance_iteree_svd.pdf b/ML1_10_puissance_iteree_svd.pdf new file mode 100644 index 0000000000000000000000000000000000000000..812006c7349e5100f8322213047a1c34452b6110 GIT binary patch literal 166844 zcma&NQ;;r95T@C-ZQJJQ{@S)}+qQMuwr%UQZQHiZ{bzS)Z)PHP@}?@QE-EjwBA$9H zlUzYWjGmc+9fo{vX?P8WorsCZ-pC5(&mR~@8B;rR7Yia5PBu=W|Jz^~#Vl=HOr3}r z#cT{+OhrtM?M+N!`1xU+U7Sn}ZDBk%u63m2a66EDKGmNTK}B?W2(91th}MAwN1zfL zkPZHY$1d^C*NSB>4`v;$mQ{|YX41b1XQtXfdOJ#^Wzk_&^?X#0F;5e)W`3WeUwi)i z>@RcfLZ>ndb1AcUzcCvtp$^NGwvbN=F8=xSbNWr|s>t^lDEHs-M~k}tY3a#w;>*D7 zY4Boj_Guh{^lR8YsnWF2ySgiZ|3TG?%JKjC9wfZ+^LTsY+;z<2K44odAqsIqN;IQm z{j)fQT%_@G2X@e{ecR@)Onr1@`LFZPtZ}niy}-29&x^*V$bRhf{jqlKjgdS?iou~}*OYyVYjfMZ8FY@>d~ zx!=a5vfo=J*|S|pb7c}X*hCNfjt1{Fg1|a=Yoy1}v&4qbsE__LTGmCKb<_oKWvw=J zFtu8xT2Xt4;s&DVlnv1DRaY+h(+L9IH6YLQ=Ci*u@K^{{M>Qd>M#P%oGN-S zNI^wkB|B%=sCBFGo`Yi?8+b&5bLJbq!Icscl!dr}d%KuLplg5I`8)z#L)Db<21J`f zb1;FzV65wZrj$@1l3}f};wvPa3RNN2OxKTAz_T*f2!(;pG%z=g`o_p*Z=r7wgiE+u z^)>gH_C4U9O?&oy*roRy{d~AzctF zQ*HaNPXgdYhSU&;DL2d|%59pA-0;OjR4By#LaZdhjUBN(FsqxDzS10GO9Mrh3Zvoq zM(0~-s;TSj6nz3^*9=b|)EIbmtsylv7f7ebI}@8yTeR~6)`7^Nh4cVaEFROCzhEEr zez6!hkbGavq`MW93V(twP_hTbG2pOVD0xSnB;)s^TeI3M`gG7e&fyMtES+Y2je|(aP%?Mz#g23ul_|5I)KDad=N@-i|hg@ z&ejI*PuuH`0dMn_pvGbfkczKbsgs?Cd}M6xvTFKH6x_vf`bsMvVW(V*;Q#wrG@{I- zB5)xUvLpgQ>7A!m(0ElYG$wg<^qEWnjw~M4fXF9u6fH!H7+;Bh7KqH0yUg=XP6mQ% z;@Z>s-q6{#|F;4!8??^p6#9Xoz*M++*;UX_!lv?2WC1iihsT=^oAkv3WROgotR5e3 zcl}Bw4*pNt((@sAXQ)NlOc+&uS|K z86iE-F~th9qsVB}e`_WH@+~m$ivP}*6fm5V(Baz5@N>( zCxWy56E7CVn2VlR1WioYjOGGuQFcI7>ESB{t{E8&9SPd1?e-~d)Jz=;F2tCw2GQUr zcLcjftwK;N#f>-UGru>espvRkLgA1(_m%x5D^99?(|;CQAb`5D6mHdIoP$$I%!_cW z9LqBsDdTX#Mu&Wa0`>K0&z#6bKsap@G>LafAo{QUo2kCzra)kuMqsd-espNqx)Bd! z>*k;XDXRS4@4^BKC~ubL6NxL&cqTb2+vY|SU8oXGEKZ-3fo73 znwF*&(cLFR;@ObePl{i{32z!JQpv%P$xLHioIM>d$1J@@87^p#J1)KjMA$%0cn+4W z#@k6W;D?ZI5s=1VDhvA#bsX6G#R*-ZK~i2q$tlFhjP(y?^Pmc$#gHJeAH{m*dAyTe zS!b997YgEAwHmnoPsWKGiCktEvdrHw1ICs_MWh%}9xLJcFo_zdQeI>@u2R)`pS;9F zg_2qeDqAFqnSlBxxC-zAl{WNxS}nn5t^g(o?x1L5xOF~-DcMXTo^_cw1_!X6*{r9U zAa0ZLZb=V|3k3*3i+%+uczL3OM0HTr4klf;dsWt1`nv3?75Ukc>$G5?9uI6y}mn_@HF>~l0 zqORhJ)iJr0XqKd>`{bcPa~-a=!gyh$wC;%!#Nu7Wd)a`o8O&V~oru3t_8|3np9fjC zj6I4ZRE(tHrwMAL=l%TIm$m;K=?)CR<-i%gb!$H#!qJPpweMVj%Nf{ zw{`8@?`CPM@+YUN#|_8zDb##$z01#O=oe5e>q77Tu5{@KuCtY3w3O0A3$`sv{;*yv1lwC55)q{qf{>sx6Mu4a&P| z)xW=kpyH*+^zVEu#XPr%=fPfqtNuizES;0G^gaO+<-_1?AB{2sui?;9ZT`yUcbKSq z%!48AnF$;-6QuA}|H#=3FTZ@=iw}{5?V3 zSb@0j-**pgHSRfC1m8`CBoNhauqLdGbdAc>%EI)Ozwq%`p09K6qY+MnyYH<^a6G$| zrm=R)m4$~-g9d~U!BD}qO*LE?u;d2R9>3w@`ft^-xmaM7EKw>~oi0T{ z*=GVyIsl=DRBl7ovP_rAV;qAu!4jqqeb~DOu2?`SLJaMYj2^+hlJp(V6@VfKT!96Q zF}6cid@RDL^v9e!B~ZaI$40BwZ_lSsU)LSp624)@!Wp4FmPF}NW2P1lBV{H4YM5`D z5c(zvisv(Gr0IfgIwOnrs%Xkb~rbmlx2}q8N=79}) zDcVvN?I(ZGR7v7-NRffpgKW+oS}jN;rX2y<2^HEdk&1zE_?l$hWwux)ySbPRyAQ~g zrM_5#gyXCPhF^-zRgO$l1I?ds7N)e>UvauC_PBE@gIsDvkUzhLKmlilW*TK&-GLbC z1v#C|-9r(TASn}e;#c_0ziRI+Hzu z6R%D&Q&IM7;8|%hd^H7VO!kM|v9@CXr zEtSD9=IU?srH13*hwi;p6k}`>9;C&-lQxFQ(=Kc^o!r^UXD`>S>)UPZ1a;O_`#A4k zvl}RDnKd9n@6}a!U zFN72+M#LT)HuZd!IDT3-C#HX5EWJ_~>To(ICSY_N=PU{=(t`nvQYW*ssldPImr`shNpRalZ&CxLY83{}*3d=H9#t1(VeNEKZer?SLm%|$3a4jXAm-$)V*4`~V=R7mGquy2iKS~QXOet(DV2D*NJG=F$q zl;sK@1i_C$Y6QX%nxi1|PvctAiBPuSw1_fO6twWCfoql1pOT)TbWRn0jR8f(3)qq~ zAQ%3WlPM1rx@XRO=`QbF5b)oi?MF7UyROrCVXJ`~Y`cUe9#_5WCJgv+WTdjIRxJ@( zPaS7RwF+6g2!oRghMCabPQ%4dI?Y3DZ-rbVgv3ZoUjy--{Xf@8%QGFH2glo8yyozk z#PZ78NM%brXBb&sg#Lx=D7~K>!>kl1x9rChgX8`5!4|DEJ-weZyAIDdqH<)hhJ9=a zMqrNyaF1kRx$`HIV)4oJyk+p4jl#(Tf7+Zvc<|fLT=kx%VSbogo~bz)J4~V;q(-Gl zB5*Vf9X7;hmRPev;*o}tuxpFcGXe&``!4qC_b}Z&5nE;c?63?APFBk&LMy^@9b=)2 z+tzV5h!mphE?LDVJz->TOWckrM!d-|Q8qGnKC9p(p5aDc^Z%&!AGz`qwGEhRquS+h z5O%_=5q%Z^z{xvm~&(%@6Dp%^pf~^9E*f%pIA~afeHu;8(K5`7afdE^2FyjD7-O#wyefF(R!Y%gfUlV<` zy0}ZHWj;uHZEFX)0XVd%RB+T-P<<1s0qxHBAC3DG=OWd0aTM59@W)_F6uRz zel8ZoYRNE_OVrXrQK?S!g`lZ{fp{{Wke#*Wm@f0;%ZZr0L!n(dDBnuW>T!t9JFK{Cq5!Xd*2wNy?V;0A@5Xdiu==ONZrS7D%bOYQRB7QR<2Ft{ zAd;h00J4+Tz9RCVKf&e-CY(*1#Ly!rB{lrxUbdb6_`0Toe>-z#(n>!J{Bq^fZuN}f zAz${x2|e7vO1DBtz{*YU(kUqjJNx}Fq0;%7sK=(dZ+!I?7fj?wgEH5152xRr@zr|d zjtVT=s86`W&(3FM1;%}#QK&3R>EcS* zxs|yS*d*<>vpw6~Dm|*X}55p zhSx4z+}ltflp*!(hNd41yK|MM6uWz`Z(nyS!hwVz(R5U?*tGv8Z;bN&a8!JOz}h`NWb4pv9TT_6?=@MNih9Zd9`b^cL;)uMGA zIwBW0NyUDNOr}Yx4qzd+C^Fs5OgiT4lhjhBH3K~1vR7KQy7}|an~=Z@iqeAc8PoBA zc328N;c5i-aF$Pf$No-)JcL09jdU~bS0q&A^7+#27$_WRB$(GwamCs)e%uib1lavHr`?=YumkE|2*A( zY!>%@AC1K;3VHgX9rY0n1;6%72KVFY8%@eW=#n82l#Z@2q4%r!+(HOV8bZkFet&>| zwf=OpYZv!aMEaPj+pYT??7u3>_U*ywZ9aYm`!V~#rI$gB zhrrR^?BLea1_>ZS4Pg6+$0x8eb1iHMo!<-!93w*?z+nc&%NkHwK_MV3Dx(QbR8XB? 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Ainsi, nous comprendrons mieux la géométrie du SVD et nous serons plus tard en mesure de construire un algorithme qui est une extension de la méthode de la puissance itérée pour le calcul des $k$ premières valeurs singulières et vecteurs singuliers. + +# Projecteur + +Un projecteur est une matrice carrée idempotente : $\mathbf{P}^2=\mathbf{P}$. Rappelons que l'espace des colonnes de $\mathbf{P}$ formé de l'ensemble des vecteurs qui s'expriment comme une combinaison linéaire des colonnes de $\mathbf{P}$, c'est-à-dire tous les vecteurs $\mathbf{v}$ qui s'écrivent sous la forme $\mathbf{P}\mathbf{x}$. L'espace des colonnes de $\mathbf{P}$ est également appelé l'image de $\mathbf{P}$ et se note $Im(\mathbf{P})$. Un vecteur $\mathbf{v}$ qui appartient à l'image d'un projecteur $\mathbf{P}$ est tel que $\mathbf{P}\mathbf{v}=\mathbf{P}$ : + +\begin{align*} + & \mathbf{v} \text{appartient à l'image de $\mathbf{P}$} \\ += \{& \text{Par définition de l'image d'une transformation linéaire.} \} \\ + & \mathbf{v} = \mathbf{P}\mathbf{x} \\ += \{& \text{Multiplication des deux membres de l'égalité par $\mathbf{P}$} \} \\ + & \mathbf{P}\mathbf{v} = \mathbf{P^2}\mathbf{x} \\ += \{& \text{Idempotence d'un projecteur.} \} \\ + & \mathbf{P}\mathbf{v} = \mathbf{P}\mathbf{x} \\ += \{& \mathbf{v} = \mathbf{P}\mathbf{x} \} \\ + & \mathbf{P}\mathbf{v} = \mathbf{v} \\ +\end{align*} + +On appelle noyau d'une transformation linéaire, l'ensemble des vecteurs que cette dernière transforme en $\mathbf{0}$, c'est-à-dire les vecteurs qui montrent une dépendance linéaire entre des colonnes de la représentation matricielle de la transformation. Dans le cas d'un projecteur $\mathbf{P}$, la projection sur l'image de $\mathbf{P}$ d'un vecteur $\mathbf{v}$ se fait selon la direction $\mathbf{P}\mathbf{v}-\mathbf{v}$ qui appartient au noyau de $\mathbf{P}$ noté $Ker(\mathbf{P})$ (voir la figure \@ref(fig:projecteur)) : +\[ +\mathbf{P}(\mathbf{P}\mathbf{v}-\mathbf{v}) = \mathbf{P}^2\mathbf{v}-\mathbf{P}\mathbf{v} = \mathbf{P}\mathbf{v}-\mathbf{P}\mathbf{v} = \mathbf{0} +\] + +```{r projecteur, fig.cap= "Schéma d'un projecteur", echo=FALSE} +knitr::include_graphics("images/projecteur1.jpg") +``` + +Le projecteur complémentaire à $\mathbf{P}$ est $\mathbf{I-P}$, il projette sur le noyau de $\mathbf{P}$. C'est-à-dire que $Im(\mathbf{I-P})=Ker(\mathbf{P})$. En effet : + +\begin{align*} + & \mathbf{v} \in Ker(\mathbf{P}) \\ += \{& \text{Par définition du noyau} \} \\ + & \mathbf{P}\mathbf{v} = \mathbf{0} \\ +\Rightarrow \phantom{\{}& \\ + & (\mathbf{I-P})\mathbf{v} = \mathbf{v} \\ += \{& \text{Par définition de l'image} \} \\ + & \mathbf{v} \in Im(\mathbf{I-P}) \\ +\end{align*} + +Donc, $Ker(\mathbf{P}) \subseteq Im(\mathbf{I-P})$. Par ailleurs, un élément de l'image de $\mathbf{I-P}$ peut se noter $(\mathbf{I-P})\mathbf{v}$, ce qui est égal à $\mathbf{v}-\mathbf{Pv}$ et appartient donc au noyau de $\mathbf{P}$. Donc, nous avons aussi $Im(\mathbf{I-P}) \subseteq Ker(\mathbf{P})$. Ainsi : $Ker(\mathbf{P}) = Im(\mathbf{I-P})$ (de même : $Ker(\mathbf{I-P}) = Im(\mathbf{P})$). + +Donc, un projecteur $\mathbf{P}$ sépare l'espace initial en deux sous-espaces $S1 = Im(\mathbf{P})$ et $S2 = Ker(\mathbf{P})$. $\mathbf{P}$ projette **sur** $S1$ **parallèlement à** (ou **le long de**) $S2$. + +# Projecteur orthogonal + +Un projecteur est dit orthogonal quand $S1$ et $S2$ sont orthogonaux. Montrons qu'une projection $\mathbf{P}$ est orthogonale si et seulement si $\mathbf{P}=\mathbf{P}^T$. Montrons d'abord que $\mathbf{P}=\mathbf{P}^T \Rightarrow \text{orthogonalité}$. + +\begin{align*} + & (\mathbf{Px})^T(\mathbf{I-P})\mathbf{y} \quad \text{pour tout $\mathbf{x}$ et $\mathbf{y}$} \\ += \{& \text{Propriété de la transposition.} \} \\ + & \mathbf{x}^T\mathbf{P}^T(\mathbf{I-P})\mathbf{y} \\ += \{& \mathbf{P}=\mathbf{P}^T \} \\ + & \mathbf{x}(\mathbf{P}-\mathbf{P}^2)\mathbf{y} \\ += \{& \mathbf{P} \text{ est un projecteur donc } \mathbf{P}=\mathbf{P}^2\} \\ + & 0 +\end{align*} + +Montrons ensuite que $\text{orthogonalité} \Rightarrow \mathbf{P}=\mathbf{P}^T$. $\mathbf{P}$ projette sur $S1$ le long de $S2$ avec $S1 \bot S2$. Soit $\{\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_n}\}$ une base orthonormale de $S1$ et $\{\mathbf{q_{n+1}}, \mathbf{q_{n+2}}, \dots, \mathbf{q_m}\}$ une base de $S2$. Soit $Q$ la matrice orthogonale dont la jème colonne est $\mathbf{q_j}$. Nous avons : $\mathbf{PQ} = (\mathbf{q_1}, \mathbf{q_2}, \dots, \mathbf{q_n}, 0, \dots, 0)$. Donc, $\mathbf{Q^TPQ} = diag(1, 1, \dots, 1, 0, \dots, 0)=\mathbf{\Sigma}$. Ainsi, $\mathbf{P} = \mathbf{Q\Sigma Q^T} = \mathbf{P}^T$. + +Etant donné un vecteur unitaire $\mathbf{q}$. Le projecteur orthogonal sur l'espace à une dimension défini par $\mathbf{q}$ est la matrice de rang 1 $\mathbf{P_q}=\mathbf{q}\mathbf{q}^T$. Le projecteur complémentaire est $\mathbf{P_{\bot q}}=\mathbf{I}-\mathbf{P_q}$ de rang $m-1$ (si $\mathbf{q}$ est de dimension $m$). + +Pour toute matrice $\mathbf{Q} \in \mathbb{R}^{M \times N}$ dont les colonnes $\mathbf{q_j}$ sont orthonormales, $\mathbf{P}=\mathbf{QQ^T}=\sum \mathbf{q_j}\mathbf{q_j}^T$ est un projecteur orthogonal sur $Im(\mathbf{Q})$. + +Rappelons la décomposition en valeurs singulière d'une matrice $\mathbf{A} \in \mathbb{R}^{M \times N}$ de rang $r$ : $\mathbf{A} = \mathbf{U} \mathbf{\Sigma}\mathbf{V}^T$ avec $\mathbf{U} \in \mathbb{R}^{M \times M}$, $\mathbf{V} \in \mathbb{R}^{N \times N}$ des matrices aux colonnes orthonormales et $\mathbf{\Sigma}=diag(\sigma_1,\dots,\sigma_r) \in \mathbb{R}^{M \times N}$. Nous pouvons partitionner le SVD en blocs pour obtenir la forme réduite présentée sur la figure \@ref(fig:svdreduit). + +```{r svdreduit, fig.cap= "SVD réduit", echo=FALSE} +knitr::include_graphics("images/SVD_reduit.jpg") +``` + +Nous avons les propriété fondamentales suivantes : + +\begin{align*} +Im(\mathbf{A}) &= Im(\mathbf{U_r}) & \mathbf{U_r}\mathbf{U_r}^T \text{ est un projecteur orthogonal sur } Im(\mathbf{A}) \\ +Ker(\mathbf{A}) &= Im(\tilde{\mathbf{V_r}}^T) & \tilde{\mathbf{V_r}}\tilde{\mathbf{V_r}}^T \text{ est un projecteur orthogonal sur } Ker(\mathbf{A}) \\ +\end{align*} \ No newline at end of file diff --git a/ML1_11_projecteurs.pdf b/ML1_11_projecteurs.pdf new file mode 100644 index 0000000000000000000000000000000000000000..706eea8b8ad39ef6fbdc97fa286e4438c12a242a GIT binary patch literal 334018 zcmeFYcT|&Iw=WucZ_;aM(t8(>fJhS%6r~D5igcsWTSD(eK)N(RPJ+%htXO5A@pa!G9$cQ3~`+>+Yv&%GS)JK94W9Ld$x 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z41qucB_ysQz^)za5LXu09l+}m6lf+bEH0=Z4^a|S77>wGk`RX~3MfDYp;tD-Lc&ma z1rb@`|7~)WMicI4k9GhGiHiQGv-AI8#&HZc@P+D9x9_1^{pD=vSD5-HjsxdXsshM9 z#5R!iXJ0SsY$SIE2$0KQl~?5KRs9G7hC!3*}o2XI%@EU5mgY z=lgjSBTqhycppC6WnuVnqnxiyd2}Yg_!Xu}u8vXfIH`)!C?~p%bJRl4SPK&z{l0KC z_##KU^itcT|Iy7OOE!x6JFkV&k!?Ezvc3GxPw%Pz&x)haRvu^{kE;?xAP`{*F#rdL J(qm=7e*sBO-}e9j literal 0 HcmV?d00001 diff --git a/ML1_regression.Rmd b/ML1_regression.Rmd index 1756329..3f5ee94 100644 --- a/ML1_regression.Rmd +++ b/ML1_regression.Rmd @@ -365,20 +365,20 @@ Nous présentons une méthode générique de compression d'un tableau rectangula Soit un tableau de données $\mathbf{X}$ composé de $N$ observations en lignes, chacune décrite par $P$ variables en colonnes. $x_{ij}$ est la valeur de la variable $j$ pour l'observation $i$. Les vecteurs lignes forment $N$ points de l'espace $\mathbb{R}^P$. Les vecteurs colonnes forment $P$ points de l'espace $\mathbb{R}^N$. Nous cherchons à projeter le nuage des points lignes sur un sous-espace $\mathcal{H} \subset \mathbb{R}^P$, tout en minimisant les déformations. -Pour commencer, nous considèrons le meilleur sous-espace $\mathcal{H}$ à une dimension, c'est-à-dire une droite définie par son vecteur directeur unitaire $\mathbf{v}$ ("unitaire" signifie que sa norme est égale à $1$, soit $\mathbf{v}^T\mathbf{v}=1$). Soit $M_i$ un des $N$ points de $\mathbb{R}^P$. A ce point correspond le vecteur $\mathbf{OM_i}$ aussi noté $\mathbf{x_i}$ car ses coordonnées sont données par la i-ème ligne de $\mathbf{X}$. Soit $H_i$ la projection de $M_i$ sur la droite $\mathcal{H}$. +Pour commencer, nous considèrons le meilleur sous-espace $\mathcal{H}$ à une dimension, c'est-à-dire une droite définie par son vecteur directeur unitaire $\mathbf{v}$ ("unitaire" signifie ici que sa norme euclidienne est égale à $1$, soit $\sqrt{\mathbf{v}^T\mathbf{v}}=1$, autrement dit, $\mathbf{v}^T\mathbf{v}=1$). Soit $M_i$ un des $N$ points de $\mathbb{R}^P$. A ce point correspond le vecteur $\mathbf{OM_i}$ aussi noté $\mathbf{x_i}$ car ses coordonnées se lisent sur la i-ème ligne de $\mathbf{X}$. Soit $H_i$ la projection de $M_i$ sur la droite $\mathcal{H}$. ![svd1](images/svd1.jpeg) La longueur $OH_i$ est le produit scalaire de $\mathbf{x_i}$ et de $\mathbf{v}$. \[ -OH_i = \mathbf{x_i}\mathbf{v} = \sum_{j=1}^{P} x_{ij} v_j +OH_i = \mathbf{x_i}^T\mathbf{v} = \sum_{j=1}^{P} x_{ij} v_j \] -Nous obtenons un vecteur des $N$ projections $OH_i$ par le produit de la matrice $\mathbf{X}$ et du vecteur $\mathbf{v}$ : $\mathbf{X}\mathbf{v}$. En choisissant pour $\mathcal{H}$ le sous-espace qui minimise la somme des carrés des écarts, il faut minimiser $\sum_iM_iH_i^2$. +Nous obtenons un vecteur des $N$ projections $OH_i$ par le produit de la matrice $\mathbf{X}$ et du vecteur $\mathbf{v}$ : $\mathbf{X}\mathbf{v}$. Choisissons pour $\mathcal{H}$ le sous-espace qui minimise la somme des carrés des écarts entre chaque point et sa projection : $\sum_iM_iH_i^2$. Le théorème de Pythagore donne $\sum_i M_iH_i^2 = \sum_i OM_i^2 - \sum_i OH_i^2$. Puisque $\sum_i OM_i^2$ est indépendant de $\mathbf{v}$, nous devons maximiser \begin{equation} -\sum OH_i^2 = (\mathbf{X}\mathbf{v})^T(\mathbf{X}\mathbf{v}) = \mathbf{v}^T \mathbf{X}^T \mathbf{X} \mathbf{v} +\sum_i OH_i^2 = (\mathbf{X}\mathbf{v})^T(\mathbf{X}\mathbf{v}) = \mathbf{v}^T \mathbf{X}^T \mathbf{X} \mathbf{v} \label{eq:svd-quadratic} \end{equation} ...sous la contrainte $\mathbf{v}^T\mathbf{v}=1$. @@ -393,7 +393,9 @@ Nous construirons de même des vecteurs orthogonaux $\mathbf{u_1}, \mathbf{u_2}, Nous trouverons de façon similaire que $\mathbf{u_1}$ est le vecteur propre de $\mathbf{X} \mathbf{X}^T$ associé à la plus grande valeur propre $\mu_1$. De même, $\mathbf{u_2}$ est le vecteur propre de $\mathbf{X} \mathbf{X}^T$ associé à la seconde plus grande valeur propre $\mu_2$. Etc. -Autrement dit, nous pouvons écrire : +Les vecteurs $\mathbf{u_1}, \mathbf{u_2}, \dots$ et $\mathbf{v_1}, \mathbf{v_2}, \dots$ définissent ce que l'on appelle les axes principaux de la matrice des données $\mathbf{X}$. + +Nous pouvons écrire : \begin{equation} \begin{cases} @@ -417,7 +419,7 @@ Calculons la norme euclidienne, aussi appelée norme L2, de $\mathbf{X}\mathbf{v \begin{align*} & \text{Norme euclidienne de } \mathbf{X}\mathbf{v_\alpha} \\ -= \{ & \text{ Par définition de la norme L2.} \} \\ += \{ & \text{ Par définition de la norme euclidienne.} \} \\ & \sqrt{\left(\mathbf{X}\mathbf{v_\alpha}\right)^T\left(\mathbf{X}\mathbf{v_\alpha}\right)} \\ = \{ &\text{Propriété de la transposition} \} \\ & \sqrt{\mathbf{v_\alpha}^T\mathbf{X}^T\mathbf{X}\mathbf{v_\alpha}} \\ @@ -449,7 +451,13 @@ Soit $\mathbf{V}$ la matrice formée des $P$ vecteurs $\mathbf{v_\alpha}$ en col \label{eq:svd} \end{equation} -Nous avons montré que la matrice $\mathbf{X}$ peut s'écrire comme une somme de matrices de rang 1 qui sont les produits des vecteurs $\mathbf{u}$ (appelé *vecteur singulier à gauche*) et $\mathbf{v}$ (appelé *vecteur singulier à droite*) pondérés par les *valeurs singulières* $\sqrt{\lambda_\alpha}$. L'équation (\ref{eq:svd}) est celle de la *décomposition en valeurs singulières* de $\mathbf{X}$. +Nous avons montré que la matrice $\mathbf{X}$ peut s'écrire comme une somme de matrices de rang 1 qui sont les produits des vecteurs $\mathbf{u}$ (appelés *vecteurs singuliers à gauche*) et $\mathbf{v}$ (appelés *vecteurs singuliers à droite*) pondérés par les *valeurs singulières* $\sqrt{\lambda_\alpha}$. L'équation (\ref{eq:svd}) est celle de la *décomposition en valeurs singulières* de $\mathbf{X}$. + +Nous pouvons écrire ce résultat sous forme matricielle en introduisant les matrices $\mathbf{U}$, $\mathbf{V}$ et $\mathbf{D}$. Sans perte de généralité, nous exprimons les résultats pour $N>P$. La matrice $\mathbf{U}$, de dimension $N \times P$, contient en colonnes les vecteurs singuliers à gauche $\mathbf{u_1}, \mathbf{u_2}, \dots, \mathbf{u_P}$. La matrice $\mathbf{V}$, de dimension $P \times P$ contient en colonnes les vecteurs singuliers à droite $\mathbf{v_1}, \mathbf{v_2}, \dots, \mathbf{v_P}$. La matrice diagonale $\mathbf{D}$ contient sur sa diagonale les valeurs singulières $\sqrt{\lambda_1}, \sqrt{\lambda_2},\dots,\sqrt{\lambda_P}$. La décomposition en valeurs singulière de toute matrice $\mathbf{X}$ s'écrit alors : + +\[ +\mathbf{X} = \mathbf{U} \mathbf{D} \mathbf{V}^T +\] Nous obtenons une reconstitution approchée de rang $k$ de $\mathbf{X}$ en ne conservant dans l'équation (\ref{eq:svd}) que les $k$ plus grandes valeurs singulières $\sqrt{\lambda_1},\sqrt{\lambda_2}, \dots, \sqrt{\lambda_k}$ et en annulant toutes les autres. @@ -524,7 +532,7 @@ Soit $\mathbf{V}$ la matrice des vecteurs propres de $\mathbf{M}$ et $\mathbf{L} & \mathbf{M} = \mathbf{T}\mathbf{T}^T \\ \end{align*} -Ainsi, toute matrice définie non négative $\mathbf{M}$ peut être décomposée sous la forme $\mathbf{M}=\mathbf{T}\mathbf{T}^T$ où $\mathbf{T}$ est une transformation linéaire composée d'un changement d'échelle des axes du repère ($\mathbf{S}$) et d'une rotation ($\mathbf{V}$). +Ainsi, toute matrice définie non négative $\mathbf{M}$ se décompose en $\mathbf{M}=\mathbf{T}\mathbf{T}^T$ où $\mathbf{T}$ est une transformation linéaire composée d'un changement d'échelle des axes du repère ($\mathbf{S}$) et d'une rotation ($\mathbf{V}$). Nous remarquons par exemple que l'équation d'un cercle selon la métrique $\mathbf{M}$ correspond à celle d'une ellipse selon la métrique identité $\mathbf{I}$ : @@ -552,7 +560,7 @@ Enfin, l'angle formé entre $\mathbf{x}$ et $\mathbf{Mx}$ est inférieur ou éga & |\theta| \leq \pi/2 \end{align*} -Ainsi, le vecteur résultat $\mathbf{Mx}$ est contraint de rester du même côté que $\mathbf{x}$ de l'hyperplan perpendiculaire à $\mathbf{x}$ qui sépare l'espace en deux. Nous comprenons que le concept de matrice symétrique définie non négative est une généralisation pour un espace multidimensionnel du concept de nombre positif sur la droite réelle. +Ainsi, le vecteur résultat $\mathbf{Mx}$ est contraint à rester du même côté que $\mathbf{x}$ de l'hyperplan perpendiculaire à $\mathbf{x}$ qui sépare l'espace en deux. Nous comprenons alors que le concept de matrice symétrique définie non négative est une généralisation pour un espace multidimensionnel du concept de nombre positif sur la droite réelle. ## Optimisation sous contraintes par la méthode des multiplicateurs de Lagrange diff --git a/ML1_regression.pdf b/ML1_regression.pdf index fadae6932a79b2aa58f470304411ec09037b6a3e..f2466be954c39fc1eb8a8bbc575d0e1868e8a81c 100644 GIT binary patch delta 85333 zcmV)AK*YbF?}GvJ`-1_HCIL2+(P%1v$#UB`5WVLsbTfURLVy65V`j3Hr)Ehdu3X|d z&=evg!$kqfl=Jt~jfIb>{oBIIPNMAOK+I`zYVdZIPgyvShmA_!T3p0)Vx z-{-$QeHqaetP?W0!kRu|R4?jtwq}%=MSq2xL*libe!DK21Vzed_Zdc<>j0mZo?95@3$r_g3$H5)p2UG>isw!%%COzb?@n-bVrE z{?MTF@y80|kJ!;o(?kbN6I#=}R-%+bp;5xn%j+r^JBVjLjrmWE5AA3P_;4Bc$N|qR z@Mo5*)K14nT^I`1W^hAas=Aa}A9jqYtH72dB_zG zw(?Ua@Zp5qoxPS=giS_gX$11mI%Uizlh`E(olKtXb~cgZm{qP2g#^*S?I+ADYguiS z`2@b5DTO$|N18_9-qAveaZelDjP?oU70%ZPtE`aC$YK{E3ye`ZJ0hB0mW!!kc{{Q(Q+Dk&T&krwuu6Kh#0i4RLRWg^NK-W*g=qKqPNBW2GcBF3u2dE)Oez@x0 z)bhqCaKNJ>a88NmVh2aE-L3#O!4ddIm_Zi;p;u()s%Fx2k=fE8rv+{YDn6catbPp* zH6`2~sKd}K0jB_plkAJS}szzP>8I?{N2VQcBX{?39o@rixtNOBlX9cCRNg*}y zWK#83$_j%#f*N~uV=cs4{8*Li5(y=HgCgun=-J1*YsY-eP!8$*sWNG^bu%S&HMm2o zqXCi;e>`mUEpR&ydMBH1ta%*f#fg9zf}%9kWh#vHx~FjP?=qE@7#*#eT7-EUn2aTD zsI#;%X0x$OFBx84*sdYN_L0nBtj7 zYtBMfJQK29J(D$0)2Tt&h;n3=OGD5z-C_ux0ip5fn}m}$@xv&6^mQ=-Qyctm4FcA4 z&T|AVue)8gb&z*X+k+$9|8JA`!kS>H15VK4V<0S;44fr@N*e%X9nsG37t3~y;`lCM ziX2QMy;8`0i7c$rIOv#r^vN`tkG%d|noJIWLMCTCzAyfCOo_w|cUT82ce+ff8#z#j z_MQZdAAzDPM5k<_)W`}zk;t!o@pKXJ;O)ETZ~?lYlz>wadSv~?dy07@tlm^3-3k7E z;)V7S zr{w#swZHem`{+L_imm~1VN1dx{6i10wo)5`GHENHk}+CUYirnWT@6UuP03PoN}R?I zh&TBGc+I+o@+1n|tb4z@aNA7P53t7ngQiiFdLgA@&>!~VqK z`XtH6not4haPBRr66b82A+uvgrO)gmU1*jl1|a8|os;N6mGcW|9;1b#aqnhmM}e%# zuy6bs|LE)rYDn#C@C=$zPct&AH4}BlCJAJJ9FJ^|k4)_w zw7|;nJu!EVI&t=UoAVOuDgBlvQen1HJ3R@)^p=-TCdaoB$gxhnI&IW?|Lce6#M3}nY6@(??b z`(TY6tYLp1B8)y52?(oUZO+WvtU)A`F!}k@g|CDCn2eDcx_i-pChnC;;>0d;Ty~9Q zT4I~OE2kXy8K|sjM2S>+S)l8K2ZIiO3XY&P>k4U4Yd2aSwt2~hCMdi|+aK19D!`(a zuA)(8e~R2y(%;BZRLH=UIb!3C+~u)3!QU(Fr<}B>oImS!gvNGyB0H-6tS)lu8`5U$ z3K0oZA}_JOW)@;e^w5Lh8(2IYTJfitnXS_TXwGRZfp4_5w63xM?VjVl(V-PyylHaZ zf4^~*Bi{T0$E>6Yy~_*>o-g>lE*(K|RAO1i-_nBwVPBTF`K##*8B-`(D>3qLKRZOBa2d0D~VI2|Te#JMw_r>{>e*s%o>130n2PpzGHj~k4D1YTz zS(Dth5q{UNn783nX=wtypHh-7Cr&wW%2wsYn+Gn5orS_<#6zXz*QXoZO=4yz_Lh#= zj$ei(05%$Ze0^}D-EkM~{`f57>&w^AzW6#zcV$>+I@`Uz-{pFjt0;`(eD}K9{TQg| za1*Cd@cnI^s^e&;$qqdX6>-oGM}Irb_U&R0|9Ji9U6N*Dn#4fYy{HwAe(|*~JnA<| zqQbN+cQ;CfWtz$!p*q~8S_gkW0Nt^&(`34XE`qzZ>T3hr#c@y%ePd_Xy*F*kW2eIn ztQwY)9@~np;{~(v8MBW05H^gWkGUje`SDahM5{7(KY~E6N;8pr&9|fC_*N88z&JFY!51^==SHVGA=%o zsdc3pX@y`)wbnxzwr9m}X^m{j?f_~@@F|3m1&3?F>%m^#9 zz~9QubghT()SCBp#$EZ9g|6Lzat<1piCTg!#b-%%;o$4+Fj~7cH-Ca{=km}5jekL+ zT8K`BFQ;m(Z8!AREbnQ=u~�EG-;z5|4%HZ(scI!*^h1Q5+?*Z*;L?OdG!H5gt(_ z>vqn5fp~Yp%QaxuPY32$fcX{RM{Ec<;$R}c1?y&ecN(NyGy7PafI`zu6w1D~!47!N z=)0D{dd2ijzXk#``G1}oyOYAiCiN^UKQAonpGKEhVEJWrJlg(9Y}&S6rg>yjgdBJonj$y5S>f>FI$#-aasa7qN6{z?Slp?}DJue$pjCwgoT+0VLa zA}IJ`BuKZjspn;6JTtwdr3)iIAb0+27wZ35>mLGu zZ!0Q3EzQw8#MA_?ex4I+xC#;+;9Z?!8zxpz&0^`L0A_0^K9|NI6{j3W1G5v$>N0colc9!c41O62)PTh=fJz3s6;=1_|w+0i&*{!Jtc%@(U5UD zlk>2QedRMjfjAGeLM3!eWO6F0iDn*fiChG4mkKn1bI1+%EW${8htJEfNK!hys4^E- z0u|J`D=mj{mL?U*n~=a+&MJs%d!^xJu8@aoU7f0WZK}WmZgL?dw`&7C)hGx(Q7SOg z#Bt7Yvldqo53YAma__*sOm%M!FD(;|(cWmPb~5Xb->$)PXL(qZbZjh|Rc*JLCpr9QkmEHsK;BhHoVgs#ZLY{b+}P1uT%@>yeLqDiq?K9(mrSi$VZ*covQK%K>DDg>Y~iJkx)2qk zk`-LF$6>VdL$@)Mak-b_k!D*jXk84D88-;CU^L{s!$MuLkvt`n<7W~%08v%&Ugqe! z_7Z5DaX=T6-G?oo&b%0hH?0JLESLs=q{|c*X1Et_VACtU9qCZTtoAn!(R7K{3WVf= z6WLh|l>!)9DQ?KiIGn;b+XnH1BRay5Z+LecU92?{f~kS(b^*ciD4Cz*qoQj~ZATSI zLRXUD@cPE}r8Q%%oAIm@#|3LT@Wyp5qN-Rzz6nH^QK$;GgcTkea7WwA6U23Y*V%W< z^x*ZpT|7Fp0#@)Npw}}Nw64}Hc4!H$y;ie_|LjCD#JtA#6$1Zr=~=Jr`R*7~s9)OI zcBZk_^VQ!I@0@uNYR!swz=gm&R#5GAYdwu);_{9)cy6oyxbS-51~9i}_O~_TaK|gXaIzOi?Z19K44JkW7NOs>H8eGwZk>V_ji-ggWbS$GDJ{M3@Z ze44H7V*1MwOkDbP5)U+g%G>*u>TR4Q&WzAb3keVS%6p2_5ILr6g3w8rXimf)kotnK zG||UJXLYG2HsQG?uW-4E=o6;73j^C{y^PT-n*HBp=PLxj@Oe6ts*Ttg@W8~|#quk6 zoFZdh;}Mf|5SBQwfNN(bPSsNPJ@v}Dl@61f4M4Ol)hP%rLIjY1piQ!j#BMQ8yTk~K z<}5>(kiS{v3|Me=YnN%FUKbMNt9S|AS_*&k>DzgVQ=bw;jNIp1$D1q844}z$P7O13 zN|n%^z!}$w>_?y|3RA^ewmm17Pf(OD1&$#j;v&sJ;%P=MB2SB* z-Y$E~l_9bW?%J$>MIkMs?88vzUhsFlGPnooLtoQhS2}aX$4pCFBPj^n+YSVJ&5~77 zglQK0Q5KQvOotjPMq-pxUUg~*Ds0eKU7x`{1`Gq8#YN$fo?F>yL>b5? 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