% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/ssvd.R
\name{ssvd}
\alias{ssvd}
\title{Sparse regularized low-rank matrix approximation.}
\usage{
ssvd(x, k = 1, n = 2, maxit = 500, tol = 0.001, center = FALSE,
  scale. = FALSE, alpha = 0, tsvd = NULL, ...)
}
\arguments{
\item{x}{A numeric real- or complex-valued matrix or real-valued sparse matrix.}

\item{k}{Matrix rank of the computed decomposition (see the Details section below).}

\item{n}{Number of nonzero components in the right singular vectors. If \code{k > 1},
then a single value of \code{n} specifies the number of nonzero components
in each regularized right singular vector. Or, specify a vector of length
\code{k} indicating the number of desired nonzero components in each
returned vector. See the examples.}

\item{maxit}{Maximum number of soft-thresholding iterations.}

\item{tol}{Convergence is determined when \eqn{\|U_j - U_{j-1}\|_F < tol}{||U_j - U_{j-1}||_F < tol}, where \eqn{U_j} is the matrix of estimated left regularized singular vectors at iteration \eqn{j}.}

\item{center}{a logical value indicating whether the variables should be
shifted to be zero centered. Alternately, a centering vector of length
equal the number of columns of \code{x} can be supplied. Use \code{center=TRUE}
to perform a regularized sparse PCA.}

\item{scale.}{a logical value indicating whether the variables should be
       scaled to have unit variance before the analysis takes place.
       Alternatively, a vector of length equal the number of columns of \code{x} can be supplied.

       The value of \code{scale} determines how column scaling is performed
       (after centering).  If \code{scale} is a numeric vector with length
       equal to the number of columns of \code{x}, then each column of \code{x} is
       divided by the corresponding value from \code{scale}.  If \code{scale} is
       \code{TRUE} then scaling is done by dividing the (centered) columns of
       \code{x} by their standard deviations if \code{center=TRUE}, and the
       root mean square otherwise.  If \code{scale} is \code{FALSE}, no scaling is done.
       See \code{\link{scale}} for more details.}

\item{alpha}{Optional  scalar regularization parameter between zero and one (see Details below).}

\item{tsvd}{Optional initial rank-k truncated SVD or PCA (skips computation if supplied).}

\item{...}{Additional arguments passed to \code{\link{irlba}}.}
}
\value{
A list containing the following components:
\itemize{
   \item{u} {regularized left singular vectors with orthonormal columns}
   \item{d} {regularized upper-triangluar projection matrix so that \code{x \%*\% v == u \%*\% d}}
   \item{v} {regularized, sparse right singular vectors with columns of unit norm}
   \item{center, scale} {the centering and scaling used, if any}
   \item{lambda} {the per-column regularization parameter found to obtain the desired sparsity}
   \item{iter} {number of soft thresholding iterations}
   \item{n} {value of input parameter \code{n}}
   \item{alpha} {value of input parameter \code{alpha}}
}
}
\description{
Estimate an \eqn{{\ell}1}{l1}-penalized
singular value or principal components decomposition (SVD or PCA) that introduces sparsity in the
right singular vectors based on the fast and memory-efficient
sPCA-rSVD algorithm of Haipeng Shen and Jianhua Huang.
}
\details{
The \code{ssvd} function implements a version of an algorithm by
Shen and Huang that computes a penalized SVD or PCA that introduces
sparsity in the right singular vectors by solving a penalized least squares problem.
The algorithm in the rank 1 case finds vectors \eqn{u, w}{u, w} that minimize
\deqn{\|x - u w^T\|_F^2 + \lambda \|w\|_1}{||x - u w^T||_F^2 + lambda||w||_1}
such that \eqn{\|u\| = 1}{||u|| = 1},
and then sets \eqn{v = w / \|w\|}{v = w / ||w||} and
\eqn{d = u^T x v}{d = u^T x v};
see the referenced paper for details. The penalty \eqn{\lambda}{lambda} is
implicitly determined from the specified desired number of nonzero values \code{n}.
Higher rank output is determined similarly
but using a sequence of \eqn{\lambda}{lambda} values determined to maintain the desired number
of nonzero elements in each column of \code{v} specified by \code{n}.
Unlike standard SVD or PCA, the columns of the returned \code{v} when \code{k > 1} may not be orthogonal.
}
\note{
Our \code{ssvd} implementation of the Shen-Huang method makes the following choices:
\enumerate{
\item{The l1 penalty is the only available penalty function. Other penalties may appear in the future.}
\item{Given a desired number of nonzero elements in \code{v}, value(s) for the \eqn{\lambda}{lambda}
      penalty are determined to achieve the sparsity goal subject to the parameter \code{alpha}.}
\item{An experimental block implementation is used for results with rank greater than 1 (when \code{k > 1})
      instead of the deflation method described in the reference.}
\item{The choice of a penalty lambda associated with a given number of desired nonzero
      components is not unique. The \code{alpha} parameter, a scalar between zero and one,
      selects any possible value of lambda that produces the desired number of
      nonzero entries. The default \code{alpha = 0} selects a penalized solution with
      largest corresponding value of \code{d} in the 1-d case. Think of \code{alpha} as
      fine-tuning of the penalty.}
\item{Our method returns an upper-triangular matrix \code{d} when \code{k > 1} so
      that \code{x \%*\% v == u \%*\% d}. Non-zero
      elements above the diagonal result from non-orthogonality of the \code{v} matrix,
      providing a simple interpretation of cumulative information, or explained variance
      in the PCA case, via the singular value decomposition of \code{d \%*\% t(v)}.}
}

What if you have no idea for values of the argument \code{n} (the desired sparsity)?
The reference describes a cross-validation and an ad-hoc approach; neither of which are
in the package yet. Both are prohibitively computationally expensive for matrices with a huge
number of columns. A future version of this package will include a revised approach to
automatically selecting a reasonable sparsity constraint.

Compare with the similar but more general functions \code{SPC} and \code{PMD} in the \code{PMA} package
by Daniela M. Witten, Robert Tibshirani, Sam Gross, and Balasubramanian Narasimhan.
The \code{PMD} function can compute low-rank regularized matrix decompositions with sparsity penalties
on both the \code{u} and \code{v} vectors. The \code{ssvd} function is
similar to the PMD(*, L1) method invocation of \code{PMD} or alternatively the \code{SPC} function.
Although less general than \code{PMD}(*),
the \code{ssvd} function can be faster and more memory efficient for the
basic sparse PCA problem.
See \url{https://bwlewis.github.io/irlba/ssvd.html} for more information.

(* Note that the s4vd package by Martin Sill and Sebastian Kaiser, \url{https://cran.r-project.org/package=s4vd},
includes a fast optimized version of a closely related algorithm by Shen, Huang, and Marron, that penalizes
both \code{u} and \code{v}.)
}
\examples{

set.seed(1)
u <- matrix(rnorm(200), ncol=1)
v <- matrix(c(runif(50, min=0.1), rep(0,250)), ncol=1)
u <- u / drop(sqrt(crossprod(u)))
v <- v / drop(sqrt(crossprod(v)))
x <- u \%*\% t(v) + 0.001 * matrix(rnorm(200*300), ncol=300)
s <- ssvd(x, n=50)
table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0)
oldpar <- par(mfrow=c(2, 1))
plot(u, cex=2, main="u (black circles), Estimated u (blue discs)")
points(s$u, pch=19, col=4)
plot(v, cex=2, main="v (black circles), Estimated v (blue discs)")
points(s$v, pch=19, col=4)

# Let's consider a trivial rank-2 example (k=2) with noise. Like the
# last example, we know the exact number of nonzero elements in each
# solution vector of the noise-free matrix. Note the application of
# different sparsity constraints on each column of the estimated v.
# Also, the decomposition is unique only up to sign, which we adjust
# for below.
set.seed(1)
u <- qr.Q(qr(matrix(rnorm(400), ncol=2)))
v <- matrix(0, ncol=2, nrow=300)
v[sample(300, 15), 1] <- runif(15, min=0.1)
v[sample(300, 50), 2] <- runif(50, min=0.1)
v <- qr.Q(qr(v))
x <- u \%*\% (c(2, 1) * t(v)) + .001 * matrix(rnorm(200 * 300), 200)
s <- ssvd(x, k=2, n=colSums(v != 0))

# Compare actual and estimated vectors (adjusting for sign):
s$u <- sign(u) * abs(s$u)
s$v <- sign(v) * abs(s$v)
table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0)
table(actual=v[, 2] != 0, estimated=s$v[, 2] != 0)
plot(v[, 1], cex=2, main="True v1 (black circles), Estimated v1 (blue discs)")
points(s$v[, 1], pch=19, col=4)
plot(v[, 2], cex=2, main="True v2 (black circles), Estimated v2 (blue discs)")
points(s$v[, 2], pch=19, col=4)
par(oldpar)

}
\references{
\itemize{
  \item{Shen, Haipeng, and Jianhua Z. Huang. "Sparse principal component analysis via regularized low rank matrix approximation." Journal of multivariate analysis 99.6 (2008): 1015-1034.}
  \item{Witten, Tibshirani and Hastie (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. _Biostatistics_ 10(3): 515-534.}
}
}