randomized SVD decomposes a matrix by extracting the first k singular values/vectors using k+p random projections. 

my question concerns the singular values that are output from the algorithm. why aren't the values equal to the first k-singular values if you do the full SVD? 

Below I have a simple implementation in R.

        rsvd = function(A, k=10, p=5) {
           n = nrow(A)
           y = A %*% matrix(rnorm(n * (k+p)), nrow=n)
           q = qr.Q(qr(y))
           b = t(q) %*% A
           svd = svd(b)
           list(u=q %*% svd$u, d=svd$d, v=svd$v)
           }
    
    > set.seed(10)
    
    > A <- matrix(rnorm(500*500),500,500)
    
    > svd(A)$d[1:15]
     [1] 44.94307 44.48235 43.78984 43.44626 43.27146 43.15066 42.79720 42.54440 42.27439 42.21873 41.79763 41.51349 41.48338 41.35024 41.18068
    
    > rsvd.o(A,10,5)$d
     [1] 34.83741 33.83411 33.09522 32.65761 32.34326 31.80868 31.38253 30.96395 30.79063 30.34387 30.04538 29.56061 29.24128 29.12612 27.61804