SVD in a term document matrix do not give me values I want

Question

I am trying to replicate an example in a paper called "An introduction to LSA": An introduction to LSA

In the example they have the following term-document matrix:

And then they apply SVD and get the following:

Trying to replicate this, I wrote the following R code:

library(lsa); library(tm)

d1 = "Human machine interface for ABC computer applications"
d2 = "A survey of user opinion of computer system response time"
d3 = "The EPS user interface management system"
d4 = "System and human system engineering testing of EPS"
d5 <- "Relation of user perceived response time to error measurement"
d6 <- "The generation of random, binary, ordered trees"
d7 <- "The intersection graph of paths in trees"
d8 <- "Graph minors IV: Widths of trees and well-quasi-ordering"
d9 <- "Graph minors: A survey"

# Words that appear in at least two of the titles
D <- c(d1, d2, d3, d4, d5, d6, d7, d8, d9)

corpus <- Corpus(VectorSource(D))

# Remove Punctuation
corpus <- tm_map(corpus, removePunctuation)

# tolower
corpus <- tm_map(corpus, content_transformer(tolower))

# Stopword Removal
corpus <- tm_map(corpus, function(x) removeWords(x, stopwords("english")))

# term document Matrix
myMatrix <- TermDocumentMatrix(corpus)

# Delete terms that only appear in a document
rowTotals <- apply(myMatrix, 1, sum)
myMatrix.new <- myMatrix[rowTotals > 1, ]

# Correlation Matrix of terms
cor(t(as.matrix(myMatrix.new)))

# lsaSpace <- lsa(myMatrix.new)
# myMatrix.reduced <- lsaSpace$tk %*% diag(lsaSpace$sk) %*% t(lsaSpace$dk)

mySVD <- svd(myMatrix.new)

I got the same term-document matrix, and actually obtained the same correlations:

> inspect(myMatrix.new)
<<TermDocumentMatrix (terms: 12, documents: 9)>>
Non-/sparse entries: 28/80
Sparsity           : 74%
Maximal term length: 9
Weighting          : term frequency (tf)

           Docs
Terms       1 2 3 4 5 6 7 8 9
  computer  1 1 0 0 0 0 0 0 0
  eps       0 0 1 1 0 0 0 0 0
  graph     0 0 0 0 0 0 1 1 1
  human     1 0 0 1 0 0 0 0 0
  interface 1 0 1 0 0 0 0 0 0
  minors    0 0 0 0 0 0 0 1 1
  response  0 1 0 0 1 0 0 0 0
  survey    0 1 0 0 0 0 0 0 1
  system    0 1 1 2 0 0 0 0 0
  time      0 1 0 0 1 0 0 0 0
  trees     0 0 0 0 0 1 1 1 0
  user      0 1 1 0 1 0 0 0 0
> cor(as.matrix(t(myMatrix.new)))
            computer        eps      graph      human  interface     minors
computer   1.0000000 -0.2857143 -0.3779645  0.3571429  0.3571429 -0.2857143
eps       -0.2857143  1.0000000 -0.3779645  0.3571429  0.3571429 -0.2857143
graph     -0.3779645 -0.3779645  1.0000000 -0.3779645 -0.3779645  0.7559289
human      0.3571429  0.3571429 -0.3779645  1.0000000  0.3571429 -0.2857143
interface  0.3571429  0.3571429 -0.3779645  0.3571429  1.0000000 -0.2857143
minors    -0.2857143 -0.2857143  0.7559289 -0.2857143 -0.2857143  1.0000000
response   0.3571429 -0.2857143 -0.3779645 -0.2857143 -0.2857143 -0.2857143
survey     0.3571429 -0.2857143  0.1889822 -0.2857143 -0.2857143  0.3571429
system     0.0433555  0.8237545 -0.4588315  0.4335550  0.0433555 -0.3468440
time       0.3571429 -0.2857143 -0.3779645 -0.2857143 -0.2857143 -0.2857143
trees     -0.3779645 -0.3779645  0.5000000 -0.3779645 -0.3779645  0.1889822
user       0.1889822  0.1889822 -0.5000000 -0.3779645  0.1889822 -0.3779645
            response     survey     system       time      trees       user
computer   0.3571429  0.3571429  0.0433555  0.3571429 -0.3779645  0.1889822
eps       -0.2857143 -0.2857143  0.8237545 -0.2857143 -0.3779645  0.1889822
graph     -0.3779645  0.1889822 -0.4588315 -0.3779645  0.5000000 -0.5000000
human     -0.2857143 -0.2857143  0.4335550 -0.2857143 -0.3779645 -0.3779645
interface -0.2857143 -0.2857143  0.0433555 -0.2857143 -0.3779645  0.1889822
minors    -0.2857143  0.3571429 -0.3468440 -0.2857143  0.1889822 -0.3779645
response   1.0000000  0.3571429  0.0433555  1.0000000 -0.3779645  0.7559289
survey     0.3571429  1.0000000  0.0433555  0.3571429 -0.3779645  0.1889822
system     0.0433555  0.0433555  1.0000000  0.0433555 -0.4588315  0.2294157
time       1.0000000  0.3571429  0.0433555  1.0000000 -0.3779645  0.7559289
trees     -0.3779645 -0.3779645 -0.4588315 -0.3779645  1.0000000 -0.5000000
user       0.7559289  0.1889822  0.2294157  0.7559289 -0.5000000  1.0000000

However I tried to apply SVD to the matrix, and the only values that are equal are the eigenvalues, I cannot get what they got in the paper.

> mySVD
$d
[1] 3.3408838 2.5417010 2.3539435 1.6445323 1.5048316 1.3063820 0.8459031
[8] 0.5601344 0.3636768

$u
             [,1]        [,2]       [,3]          [,4]        [,5]        [,6]
 [1,] -0.24047023 -0.04315195  0.1644291  0.5949618181 -0.10675529 -0.25495513
 [2,] -0.30082816  0.14127047 -0.3303084 -0.1880919179  0.11478462  0.27215528
 [3,] -0.03613585 -0.62278523 -0.2230864 -0.0007000721 -0.06825294  0.11490895
 [4,] -0.22135078  0.11317962 -0.2889582  0.4147507404 -0.10627512 -0.34098332
 [5,] -0.19764540  0.07208778 -0.1350396  0.5522395837  0.28176894  0.49587801
 [6,] -0.03175633 -0.45050892 -0.1411152  0.0087294706 -0.30049511  0.27734340
 [7,] -0.26503747 -0.10715957  0.4259985 -0.0738121922  0.08031938 -0.16967639
 [8,] -0.20591786 -0.27364743  0.1775970  0.0323519366 -0.53715000  0.08094398
 [9,] -0.64448115  0.16730121 -0.3611482 -0.3334616013 -0.15895498 -0.20652259
[10,] -0.26503747 -0.10715957  0.4259985 -0.0738121922  0.08031938 -0.16967639
[11,] -0.01274618 -0.49016179 -0.2311202 -0.0248019985  0.59416952 -0.39212506
[12,] -0.40359886 -0.05707026  0.3378035 -0.0991137295  0.33173372  0.38483192
              [,7]          [,8]        [,9]
 [1,] -0.302240236  0.0623280150 -0.49244436
 [2,]  0.032994110 -0.0189980144  0.16533917
 [3,]  0.159575477 -0.6811254380 -0.23196123
 [4,]  0.522657771 -0.0604501376  0.40667751
 [5,] -0.070423441 -0.0099400372  0.10893027
 [6,]  0.339495286  0.6784178789 -0.18253498
 [7,]  0.282915727 -0.0161465472  0.05387469
 [8,] -0.466897525 -0.0362988295  0.57942611
 [9,] -0.165828575  0.0342720233 -0.27069629
[10,]  0.282915727 -0.0161465472  0.05387469
[11,] -0.288317461  0.2545679452  0.22542407
[12,]  0.002872175 -0.0003905042 -0.01232935

$v
              [,1]        [,2]        [,3]        [,4]        [,5]          [,6]
 [1,] -0.197392802  0.05591352 -0.11026973  0.94978502  0.04567856 -7.659356e-02
 [2,] -0.605990269 -0.16559288  0.49732649  0.02864890 -0.20632728 -2.564752e-01
 [3,] -0.462917508  0.12731206 -0.20760595 -0.04160920  0.37833623  7.243996e-01
 [4,] -0.542114417  0.23175523 -0.56992145 -0.26771404 -0.20560471 -3.688609e-01
 [5,] -0.279469108 -0.10677472  0.50544991 -0.15003543  0.32719441  3.481305e-02
 [6,] -0.003815213 -0.19284794 -0.09818424 -0.01508149  0.39484121 -3.001611e-01
 [7,] -0.014631468 -0.43787488 -0.19295557 -0.01550719  0.34948535 -2.122014e-01
 [8,] -0.024136835 -0.61512190 -0.25290398 -0.01019901  0.14979847  9.743417e-05
 [9,] -0.081957368 -0.52993707 -0.07927315  0.02455491 -0.60199299  3.622190e-01
             [,7]         [,8]        [,9]
 [1,]  0.17731830 -0.014393259  0.06369229
 [2,] -0.43298424  0.049305326 -0.24278290
 [3,] -0.23688970  0.008825502 -0.02407687
 [4,]  0.26479952 -0.019466944  0.08420690
 [5,]  0.67230353 -0.058349563  0.26237588
 [6,] -0.34083983  0.454476523  0.61984719
 [7,] -0.15219472 -0.761527011 -0.01797518
 [8,]  0.24914592  0.449642757 -0.51989050
 [9,]  0.03803419 -0.069637550  0.45350675

Am I missing something?

Best Regards

EDIT:

It is supposed in the example, that the dimension is reduced and they deleted the less eigenvalues. My problem is that the correlations I get after SVD are different than that of the example:

Answer 1

I managed to find my mistake. When I was reconstructing the matrix, the transpose of the M = U D V' was not computed correctly. Now it works, sorry, it was my mistake... Also, I was computing cor between documents, when what I wanted was between terms.

I added the following lines:

mySVD <- svd(myMatrix.new)

Mp <- mySVD$u[, c(1,2)] %*% diag(mySVD$d)[c(1, 2), c(1, 2)] %*% t(mySVD$v[, c(1, 2)])

rownames(Mp) <- rownames(myMatrix.new)
cor(t(Mp))