SVD of very large matrix in R

Question

I want to generate a matrix of 30000 x 30000 in r, multiplying a vector of 30000 elements by its transpose and then obtain SVD of that matrix, but the program tells me that r can not locate a vector of size 900000000. Help me, what can I do?

y <- read.csv("C:\\Users\\jmarescr\\Desktop\\BigLetra50.csv",header=TRUE)

x <- matrix(y[1:30000,1],30000,1)
tx <- as.matrix(t(x))

mat <- x %*% tx

Error: can not allocate vector of length 900000000

s <- svd(mat)

Error in svd (x): object 'mat' not found

Answer 1

Part of the beauty of the SVD is that you don't need to take the crossproduct of x to get the crossproduct's SVD.

You can, instead, get the SVD of x%*%t(x) (aka tcrossprod(x) ) directly from the elements of x 's SVD. Specifically (and up to the sign of U's columns) SVD(x %*% t(x)) = UD^2 t(U), where U and D are taken from the SVD of x . (For a reference, see here .)

To see it in action, try out a smaller example:

set.seed(1)
x <- matrix(rnorm(15), ncol=5)


svd(x)$d
# [1] 3.046842 1.837539 1.411585
sqrt(svd(tcrossprod(x))$d)
# [1] 3.046842 1.837539 1.411585

svd(x)$u
#            [,1]       [,2]      [,3]
# [1,] -0.3424029  0.7635281 0.5475264
# [2,] -0.8746155 -0.4719093 0.1111273
# [3,]  0.3432316 -0.4408248 0.8293766
svd(tcrossprod(x))$u
#            [,1]       [,2]      [,3]
# [1,] -0.3424029  0.7635281 0.5475264
# [2,] -0.8746155 -0.4719093 0.1111273
# [3,]  0.3432316 -0.4408248 0.8293766
svd(tcrossprod(x))$v
#            [,1]       [,2]      [,3]
# [1,] -0.3424029  0.7635281 0.5475264
# [2,] -0.8746155 -0.4719093 0.1111273
# [3,]  0.3432316 -0.4408248 0.8293766

Another way to see this:

sss <- svd(x)

with(sss, u %*% diag(d)^2 %*% t(u))
#           [,1]      [,2]      [,3]
# [1,]  3.654154  1.684675 -1.322649
# [2,]  1.684675  7.877802 -1.900721
# [3,] -1.322649 -1.900721  3.120415

tcrossprod(x)
#           [,1]      [,2]      [,3]
# [1,]  3.654154  1.684675 -1.322649
# [2,]  1.684675  7.877802 -1.900721
# [3,] -1.322649 -1.900721  3.120415