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\name{nmf_update.KL.h}
\alias{nmf_update.KL}
\alias{nmf_update.KL.h}
\alias{nmf_update.KL.h_R}
\alias{nmf_update.KL.w}
\alias{nmf_update.KL.w_R}
\title{NMF Multiplicative Updates for Kullback-Leibler Divergence}
\usage{
nmf_update.KL.h(v, w, h, nbterms = 0L, ncterms = 0L,
copy = TRUE)
nmf_update.KL.h_R(v, w, h, wh = NULL)
nmf_update.KL.w(v, w, h, nbterms = 0L, ncterms = 0L,
copy = TRUE)
nmf_update.KL.w_R(v, w, h, wh = NULL)
}
\arguments{
\item{v}{target matrix}
\item{w}{current basis matrix}
\item{h}{current coefficient matrix}
\item{nbterms}{number of fixed basis terms}
\item{ncterms}{number of fixed coefficient terms}
\item{copy}{logical that indicates if the update should
be made on the original matrix directly (\code{FALSE}) or
on a copy (\code{TRUE} - default). With \code{copy=FALSE}
the memory footprint is very small, and some speed-up may
be achieved in the case of big matrices. However, greater
care should be taken due the side effect. We recommend
that only experienced users use \code{copy=TRUE}.}
\item{wh}{already computed NMF estimate used to compute
the denominator term.}
}
\value{
a matrix of the same dimension as the input matrix to
update (i.e. \code{w} or \code{h}). If \code{copy=FALSE},
the returned matrix uses the same memory as the input
object.
}
\description{
Multiplicative updates from \cite{Lee et al. (2001)} for
standard Nonnegative Matrix Factorization models \eqn{V
\approx W H}, where the distance between the target
matrix and its NMF estimate is measured by the
Kullback-Leibler divergence.
\code{nmf_update.KL.w} and \code{nmf_update.KL.h} compute
the updated basis and coefficient matrices respectively.
They use a \emph{C++} implementation which is optimised
for speed and memory usage.
\code{nmf_update.KL.w_R} and \code{nmf_update.KL.h_R}
implement the same updates in \emph{plain R}.
}
\details{
The coefficient matrix (\code{H}) is updated as follows:
\deqn{ H_{kj} \leftarrow H_{kj} \frac{\left( sum_i
\frac{W_{ik} V_{ij}}{(WH)_{ij}} \right)}{ sum_i W_{ik} }.
}{ H_kj <- H_kj ( sum_i [ W_ik V_ij / (WH)_ij ] ) / (
sum_i W_ik ) }
These updates are used in built-in NMF algorithms
\code{\link[=KL-nmf]{KL}} and
\code{\link[=brunet-nmf]{brunet}}.
The basis matrix (\code{W}) is updated as follows: \deqn{
W_{ik} \leftarrow W_{ik} \frac{ sum_j [\frac{H_{kj}
A_{ij}}{(WH)_{ij}} ] }{sum_j H_{kj} } }{ W_ik <- W_ik (
sum_u [H_kl A_il / (WH)_il ] ) / ( sum_l H_kl ) }
}
\author{
Update definitions by \cite{Lee2001}.
C++ optimised implementation by Renaud Gaujoux.
}
\references{
Lee DD and Seung H (2001). "Algorithms for non-negative
matrix factorization." _Advances in neural information
processing systems_. <URL:
http://scholar.google.com/scholar?q=intitle:Algorithms+for+non-negative+matrix+factorization\#0>.
}
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