# randSVD

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##### Randomized Singular Value Decomposition.

Compute the near-optimal low-rank singular value decomposition (SVD) of a rectangular matrix. The algorithm follows a randomized approach.

##### Usage
randSVD(A, k = NULL, l = NULL, its = 2, sdist = "unif")
##### Arguments
A

array_like a real/complex input matrix (or data frame), with dimensions $(m, n)$. It is the real/complex matrix being approximated.

k

int, optional determines the target rank of the low-rank decomposition and should satisfy $k << min(m,n)$. Set by default to 6.

l

int, optional block size of the block Lanczos iterations; $l$ must be a positive integer greater than $k$, and defaults $l=k+2$.

its

int, optional number of full iterations of a block Lanczos method to conduct; $its$ must be a nonnegative integer, and defaults to 2.

sdist

str $c('normal', 'unif')$, optional Specifies the sampling distribution. $'unif'$ : (default) Uniform [-1,1]. $'normal$' : Normal ~N(0,1).

##### Details

Randomized SVD (randSVD) is a fast algorithm to compute the approximate low-rank SVD of a rectangular $(m,n)$ matrix $A$ using a probabilistic algorithm. Given the decided rank $k << n$, rSVD factors the input matrix $A$ as $A = U * diag(S) * V'$, which is the typical SVD form. Precisely, the columns of U are orthonormal, as are the columns of V, the entries of S are all nonnegative, and the only nonzero entries of S appear in non-increasing order on its diagonal. The dimensions are: U is $(m,k)$, V is $(n,k)$, and S is $(k,k)$, when A is $(m,n)$.

Increasing $its$ or $l$ improves the accuracy of the approximation USV' to A.

The parameter $its$ specifies the number of normalized power iterations (subspace iterations) to reduce the approximation error. This is recommended if the the singular values decay slowly. In practice 1 or 2 iterations achieve good results, however, computing power iterations increases the computational time. The number of power iterations is set to $its=2$ by default.

##### Value

randSVD returns a list containing the following three components:

d

array_like $(k,k)$ matrix in the rank-k approximation USV' to A, where A is $(m,n)$; the entries of $S$ are all nonnegative, and its only nonzero entries appear in nonincreasing order on the diagonal.

u

matrix $(m, k)$ matrix in the rank-$k$ approximation $A = U * diag(S) * V'$ to A; the columns of U are orthonormal and are called Left singular vect. We want to remark that this is the transpose matrix, hence the vectors are on the rows of our matrix.

v

matrix $(n, k)$ matrix in the rank-$k$ approximation $A = U * diag(S) * V'$ to A; the columns of V are orthonormal and are called Right singular vect.

##### Note

The singular vectors are not unique and only defined up to sign (a constant of modulus one in the complex case). If a left singular vector has its sign changed, changing the sign of the corresponding right vector gives an equivalent decomposition.

##### References

• [1] N. Halko, P. Martinsson, and J. Tropp. "Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions" (2009). (available at arXiv http://arxiv.org/abs/0909.4061).

• [2] S. Voronin and P.Martinsson. "RSVDPACK: Subroutines for computing partial singular value decompositions via randomized sampling on single core, multi core, and GPU architectures" (2015). (available at arXiv http://arxiv.org/abs/1502.05366).

• [3] N. Benjamin Erichson. "Randomized Singular Value Decomposition (rsvd): R package" (2016). (available in the CRAN).

• [4] Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp. "Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions" (2009). (available at http://arxiv.org).

• [5] V. Rokhlin, A. Szlam, M. Tygert. "A randomized algorithm for principal component analysis" (2009). (available at https://arxiv.org/abs/0809.2274). The implementation of rand SVD is inspired by the MatLab implementation of RandPCA by M. Tygert.

• randSVD
##### Examples
# NOT RUN {
#Simulate a general matrix with 1000 rows and 1000 columns
vy= rnorm(1000*1000,0,1)
y= matrix(vy,1000,1000,byrow=TRUE)

#Compute the randSVD for the first hundred components of the matrix y and measure the time
start.time <- Sys.time()
prova1= randSVD(y,k=100)
Sys.time()- start.time

#Compare with a classical SVD
start.time <- Sys.time()
prova2= svd(y)
Sys.time()- start.time

# }
`
Documentation reproduced from package RGeode, version 0.1.0, License: GPL (>= 2)

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