1 Introducción

Si tenemos una matriz \(A \in \mathbb{R}^{m \times n}\) (cuadrada o no, simetrica o no) tiene una factorización de la forma:

\[ A = U \Sigma V^T \] donde \(U \in \mathbb{R}^{m \times n}\) con columnas ortogonales y \(V \in \mathbb{R}^{n \times n}\) matriz ortogonal y D una matriz diagonal\((n \times n)\). Este resultado se llama descomposición en valores singulares”(DVS), y es una de las más importantes entre las descomposiciones de matrices. Además, \(( \Sigma )\) contiene los valores singulares, que se obtienen como:

\[ \sigma_i = \sqrt{\lambda_i(A^T A)} \] Ahora, explicaré como obtenerla considerando datos simulados, y con una amplicación de rango \(1\).

2 Simulación de datos

En lugar de usar una matriz fija, generamos una matriz aleatoria controlada.

set.seed(123)  # Reproducibilidad

# Dimensiones
m <- 6   # número de filas
n <- 4   # número de columnas

# Simulación de matriz con estructura
A <- matrix(rnorm(m * n, mean = 0, sd = 1), nrow = m)

A

##             [,1]       [,2]       [,3]       [,4]
## [1,] -0.56047565  0.4609162  0.4007715  0.7013559
## [2,] -0.23017749 -1.2650612  0.1106827 -0.4727914
## [3,]  1.55870831 -0.6868529 -0.5558411 -1.0678237
## [4,]  0.07050839 -0.4456620  1.7869131 -0.2179749
## [5,]  0.12928774  1.2240818  0.4978505 -1.0260044
## [6,]  1.71506499  0.3598138 -1.9666172 -0.7288912

3 Paso 1: Calcular \(A^TA\)

AtA <- t(A) %*% A
AtA

##           [,1]       [,2]       [,3]       [,4]
## [1,]  5.759821 -0.2938070 -4.2990114 -3.3468073
## [2,] -0.293807  4.1110473 -0.4680829  0.2337788
## [3,] -4.299011 -0.4680829  7.7903245  1.3554449
## [4,] -3.346807  0.2337788  1.3554449  3.4871599

4 Paso 2: Autovalores y autovectores

eig <- eigen(AtA)

lambda <- eig$values
V <- eig$vectors

lambda

## [1] 12.3469903  4.6430094  3.4993672  0.6589858

##              [,1]       [,2]       [,3]       [,4]
## [1,]  0.629272015  0.2771954 -0.2420838 0.68452529
## [2,]  0.007386372 -0.7356815 -0.6752641 0.05231205
## [3,] -0.696811267  0.4511279 -0.4767018 0.28929749
## [4,] -0.344116596 -0.4223968  0.5081035 0.66707986

5 Paso 3: Valores singulares

Los valores singulares dependen de los autovalores de \(A^TA\)

\[\begin{equation} A^T A v_i = \lambda_i v_i \end{equation}\]

Son las raíces cuadradas de los autovalores.

\[\begin{equation} \sigma_i = \sqrt \lambda_i \end{equation}\]

sigma <- sqrt(lambda)
sigma

## [1] 3.5138284 2.1547643 1.8706596 0.8117794

6 Paso 4: Construcción de \(\Sigma\)

\[\begin{equation} \Sigma = \begin{pmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \\ 0 & 0 & \sigma_3 \end{pmatrix} \end{equation}\]

Sigma <- diag(sigma)
Sigma

##          [,1]     [,2]    [,3]      [,4]
## [1,] 3.513828 0.000000 0.00000 0.0000000
## [2,] 0.000000 2.154764 0.00000 0.0000000
## [3,] 0.000000 0.000000 1.87066 0.0000000
## [4,] 0.000000 0.000000 0.00000 0.8117794

7 Paso 5: Construcción de \(U\)

Se obtiene usando los autovectores de \(A^TA\)

\[\begin{equation} u_i = \frac{Av_i}{\sigma_i} \end{equation}\]

U <- A %*% V %*% diag(1 / sigma)
U

##             [,1]        [,2]         [,3]       [,4]
## [1,] -0.24756400 -0.28304723 -0.005476774  0.2762503
## [2,] -0.01952801  0.51816137  0.329821171 -0.6246890
## [3,]  0.49249712  0.52797492 -0.102170103  0.1945325
## [4,] -0.32131770  0.57807166 -0.362817440  0.4884259
## [5,]  0.02747886 -0.09593674 -0.864144229 -0.4777961
## [6,]  0.76927053 -0.17106970 -0.048657246  0.1695804

7.0.1 Verificación de la descomposición

\[\begin{equation} A = U \Sigma V^T \end{equation}\]

A_reconstruida <- U %*% Sigma %*% t(V)

A_reconstruida

##             [,1]       [,2]       [,3]       [,4]
## [1,] -0.56047565  0.4609162  0.4007715  0.7013559
## [2,] -0.23017749 -1.2650612  0.1106827 -0.4727914
## [3,]  1.55870831 -0.6868529 -0.5558411 -1.0678237
## [4,]  0.07050839 -0.4456620  1.7869131 -0.2179749
## [5,]  0.12928774  1.2240818  0.4978505 -1.0260044
## [6,]  1.71506499  0.3598138 -1.9666172 -0.7288912

7.0.2 Error de reconstrucción

error <- sum((A - A_reconstruida)^2)
error

## [1] 5.097937e-30

7.0.3 Uso de la función svd()

svd_A <- svd(A)

U_svd <- svd_A$u
D_svd <- svd_A$d
V_svd <- svd_A$v

U_svd

##             [,1]        [,2]         [,3]       [,4]
## [1,] -0.24756400 -0.28304723  0.005476774  0.2762503
## [2,] -0.01952801  0.51816137 -0.329821171 -0.6246890
## [3,]  0.49249712  0.52797492  0.102170103  0.1945325
## [4,] -0.32131770  0.57807166  0.362817440  0.4884259
## [5,]  0.02747886 -0.09593674  0.864144229 -0.4777961
## [6,]  0.76927053 -0.17106970  0.048657246  0.1695804

D_svd

## [1] 3.5138284 2.1547643 1.8706596 0.8117794

V_svd

##              [,1]       [,2]       [,3]       [,4]
## [1,]  0.629272015  0.2771954  0.2420838 0.68452529
## [2,]  0.007386372 -0.7356815  0.6752641 0.05231205
## [3,] -0.696811267  0.4511279  0.4767018 0.28929749
## [4,] -0.344116596 -0.4223968 -0.5081035 0.66707986

7.0.4 Reconstrucción usando svd()

Sigma_R <- diag(D_svd)

A_rec <- U_svd %*% Sigma_R %*% t(V_svd)
A_rec

##             [,1]       [,2]       [,3]       [,4]
## [1,] -0.56047565  0.4609162  0.4007715  0.7013559
## [2,] -0.23017749 -1.2650612  0.1106827 -0.4727914
## [3,]  1.55870831 -0.6868529 -0.5558411 -1.0678237
## [4,]  0.07050839 -0.4456620  1.7869131 -0.2179749
## [5,]  0.12928774  1.2240818  0.4978505 -1.0260044
## [6,]  1.71506499  0.3598138 -1.9666172 -0.7288912

8 Aproximación de bajo rango

8.0.1 Aproximación rango 1

Reducimos dimensionalidad usando el primer valor singular.

\[\begin{equation} A_1 = \sigma_1 u_1 v_1^T \end{equation}\]

u1 <- U_svd[, 1, drop = FALSE]
v1 <- V_svd[, 1, drop = FALSE]
sigma1 <- D_svd[1]

A1 <- sigma1 * u1 %*% t(v1)

A1

##             [,1]          [,2]        [,3]        [,4]
## [1,] -0.54740209 -0.0064253860  0.60615431  0.29934613
## [2,] -0.04317942 -0.0005068385  0.04781384  0.02361261
## [3,]  1.08898693  0.0127824893 -1.20586701 -0.59551111
## [4,] -0.71048290 -0.0083396226  0.78673845  0.38852666
## [5,]  0.06076000  0.0007131987 -0.06728132 -0.03322653
## [6,]  1.70097554  0.0199659895 -1.88353986 -0.93017630

8.0.2 Error de aproximación

error_rango1 <- sum((A - A1)^2)
error_rango1

## [1] 8.801362

8.0.3 Aproximación rango \(k\)

k <- 2

Uk <- U_svd[, 1:k]
Vk <- V_svd[, 1:k]
Sigmak <- diag(D_svd[1:k])

Ak <- Uk %*% Sigmak %*% t(Vk)

Ak

##              [,1]       [,2]       [,3]        [,4]
## [1,] -0.716463571  0.4422668  0.3310114  0.55696599
## [2,]  0.266313539 -0.8219067  0.5515052 -0.44800005
## [3,]  1.404341439 -0.8241740 -0.6926361 -1.07605573
## [4,] -0.365206071 -0.9247105  1.3486671 -0.13761429
## [5,]  0.003457877  0.1527941 -0.1605390  0.05409179
## [6,]  1.598797200  0.2911491 -2.0498323 -0.77447454

8.0.4 Error de aproximación rango k

error_k <- sum((A - Ak)^2)
error_k

## [1] 4.158353

8.0.5 Varianza explicada

\[\begin{equation} \frac{\sigma_i ^2}{\sum \sigma_i^2} \end{equation}\]

total <- sum(D_svd^2)

var_exp <- D_svd^2 / total * 100
var_acum <- cumsum(var_exp)

data.frame(
  Componente = 1:length(D_svd),
  Varianza = var_exp,
  Acumulada = var_acum
)

##   Componente  Varianza Acumulada
## 1          1 58.382752  58.38275
## 2          2 21.954473  80.33723
## 3          3 16.546760  96.88399
## 4          4  3.116015 100.00000

Descomposición en Valores Singulares (SVD) con Datos Simulados

José Luis Vásquez Pérez