Iterative thresholding sparse SVD

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Description

The function computes sparse SVD by iterative thresholding algorithm with an initializtion as one of the inputs

Usage

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ssvd.iter.thresh(x, method = c("theory", "method"), u.old, v.old, 
gamma.u = sqrt(2), gamma.v = sqrt(2), dothres = "hard", r = ncol(u.old), 
tol = 1e-08, n.iter = 100, n.boot = 100, sigma = NA, non.orth = FALSE)

Arguments

x

Input matrix, for which one would like to get a sparse SVD.

method

If method = "theory", then a theoretical procedure is adopted which is based on normal assumption on the noise. If method = "method", then the function bypass the normal assumption by some robust statistics. These two choices typically give similar solutions, but "theory" is much faster.

u.old

A matrix that contains initial left singular vectors as the columns of the matrix.

v.old

A matrix that contains initial right singular vectors as the columns of the matrix.

gamma.u

When the method="theory", gamma.u=sqrt(2) corresponds to the sqrt(2 log(p)), which is the largest magnitude of p iid standard normals. If gamma.u is manually set to be smaller or larger than sqrt2, the left singular vectors will be denser or sparser respectively.

gamma.v

When the method="theory", gamma.u=sqrt(2) corresponds to the sqrt(2 log(p)), which is the largest magnitude of p iid standard normals. If gamma.u is manually set to be smaller or larger than sqrt2, the right singular vectors will be denser or sparser respectively.

dothres

Dothres has two choices, either "hard" or "soft", which means hard-thresholding or soft-thresholding

r

A scaler, the number of components, i.e., the number of singular vectors to be computed.

tol

The tolerance level that determines when the algorithm stops.

n.iter

Maximum number of iterations allowed.

n.boot

Number of bootstrap to estimate the threshold level when method = "method"

sigma

Sigma is a scaler for the noise level. The user can set it to be NA, and the function will estimate it automatically.

non.orth

If non.orth=TRUE, then the last iteration of the algorithm will not involve orthoganolization, which should produce sparse solutions.

Value

u

A matrix containing left singular vectors

v

A matrix containing right singular vectors

d

A vector containing singular values

niter

Number of iterations for the algorithm to converge

sigma.hat

An estimate of the noise level

dist.u

The distance between the left singular vectors of the last two iterations, can be used to see whether the algorithm indeed converges.

dist.v

The distance between the right singular vectors of the last two iterations, can be used to see whether the algorithm indeed converges.

Author(s)

Dan Yang

References

A Sparse SVD Method for High-dimensional Data

Examples

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ans.initial <- ssvd.initial(matrix(rnorm(2^15),2^7,2^8), method = "method")
ans.iter <- ssvd.iter.thresh(matrix(rnorm(2^15),2^7,2^8), 
u.old=ans.initial$u, v.old= ans.initial$v, method = "method")