Fast Singular Value Decomposition
fast.svd returns the singular value decomposition of
a rectangular real matrix
M = U D V',
where U and V are orthogonal matrices with U' U = I
and V' V = I, and D is a diagonal matrix containing the
singular values (see svd).
The main difference to the native version svd is that
fast.svd is substantially faster for "fat" (small n, large p)
and "thin" (large n, small p) matrices.
In this case the decomposition of M can be
greatly sped up by first computing the SVD of either M M' (fat matrices) or
M' M (thin matrices), rather than that of M.
fast.svd(m, tol)
m |
matrix |
tol |
tolerance - singular values larger than
tol are considered non-zero (default value:
|
For "fat" M (small n, large p) the SVD decomposition of M M' yields
M M' = U D^2 U'
As the matrix M M' has dimension n x n only, this is faster to compute than SVD of M. The V matrix is subsequently obtained by
V = M' U D^(-1)
Similarly, for "thin" M (large n, small p), the decomposition of M' M yields
M' M = V D^2 V'
which is also quick to compute as M' M has only dimension p x p. The U matrix is then computed via
U = M V D^(-1)
A list with the following components:
d |
a vector containing the positive singular values |
u |
a matrix with the corresponding left singular vectors |
v |
a matrix with the corresponding right singular vectors |
Korbinian Strimmer (https://strimmerlab.github.io).
# load corpcor library
library("corpcor")
# generate a "fat" data matrix
n = 50
p = 5000
X = matrix(rnorm(n*p), n, p)
# compute SVD
system.time( (s1 = svd(X)) )
system.time( (s2 = fast.svd(X)) )
eps = 1e-10
sum(abs(s1$d-s2$d) > eps)
sum(abs(abs(s1$u)-abs(s2$u)) > eps)
sum(abs(abs(s1$v)-abs(s2$v)) > eps)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.