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\name{csi}
\docType{methods}
\alias{csi}
\alias{csi-methods}
\alias{csi,matrix-method}
\title{Cholesky decomposition with Side Information}
\description{
The \code{csi} function in \pkg{kernlab} is an implementation of an
incomplete Cholesky decomposition algorithm which exploits side
information (e.g., classification labels, regression responses) to
compute a low rank decomposition of a kernel matrix from the data.
}
\usage{
\S4method{csi}{matrix}(x, y, kernel="rbfdot", kpar=list(sigma=0.1), rank,
centering = TRUE, kappa = 0.99 ,delta = 40 ,tol = 1e-5)
}
\arguments{
\item{x}{The data matrix indexed by row}
\item{y}{the classification labels or regression responses. In
classification y is a \eqn{m \times n} matrix where \eqn{m}
the number of data and \eqn{n} the number of classes \eqn{y} and \eqn{y_i} is 1 if
the corresponding x belongs to class i.}
\item{kernel}{the kernel function used in training and predicting.
This parameter can be set to any function, of class \code{kernel},
which computes the inner product in feature space between two
vector arguments. kernlab provides the most popular kernel functions
which can be used by setting the kernel parameter to the following
strings:
\itemize{
\item \code{rbfdot} Radial Basis kernel function "Gaussian"
\item \code{polydot} Polynomial kernel function
\item \code{vanilladot} Linear kernel function
\item \code{tanhdot} Hyperbolic tangent kernel function
\item \code{laplacedot} Laplacian kernel function
\item \code{besseldot} Bessel kernel function
\item \code{anovadot} ANOVA RBF kernel function
\item \code{splinedot} Spline kernel
\item \code{stringdot} String kernel
}
The kernel parameter can also be set to a user defined function of
class kernel by passing the function name as an argument.
}
\item{kpar}{the list of hyper-parameters (kernel parameters).
This is a list which contains the parameters to be used with the
kernel function. Valid parameters for existing kernels are :
\itemize{
\item \code{sigma} inverse kernel width for the Radial Basis
kernel function "rbfdot" and the Laplacian kernel "laplacedot".
\item \code{degree, scale, offset} for the Polynomial kernel "polydot"
\item \code{scale, offset} for the Hyperbolic tangent kernel
function "tanhdot"
\item \code{sigma, order, degree} for the Bessel kernel "besseldot".
\item \code{sigma, degree} for the ANOVA kernel "anovadot".
}
Hyper-parameters for user defined kernels can be passed through the
kpar parameter as well.
}
\item{rank}{maximal rank of the computed kernel matrix}
\item{centering}{if \code{TRUE} centering is performed (default: TRUE)}
\item{kappa}{trade-off between approximation of K and prediction of Y (default: 0.99)}
\item{delta}{number of columns of cholesky performed in advance (default: 40)}
\item{tol}{minimum gain at each iteration (default: 1e-4)}
}
\details{An incomplete cholesky decomposition calculates
\eqn{Z} where \eqn{K= ZZ'} \eqn{K} being the kernel matrix.
Since the rank of a kernel matrix is usually low, \eqn{Z} tends to
be smaller then the complete kernel matrix. The decomposed matrix can be
used to create memory efficient kernel-based algorithms without the
need to compute and store a complete kernel matrix in memory. \cr
\code{csi} uses the class labels, or regression responses to compute a
more appropriate approximation for the problem at hand considering the
additional information from the response variable. }
\value{
An S4 object of class "csi" which is an extension of the class
"matrix". The object is the decomposed kernel matrix along with
the slots :
\item{pivots}{Indices on which pivots where done}
\item{diagresidues}{Residuals left on the diagonal}
\item{maxresiduals}{Residuals picked for pivoting}
\item{predgain}{predicted gain before adding each column}
\item{truegain}{actual gain after adding each column}
\item{Q}{QR decomposition of the kernel matrix}
\item{R}{QR decomposition of the kernel matrix}
slots can be accessed either by \code{object@slot}
or by accessor functions with the same name
(e.g., \code{pivots(object))}}
\references{
Francis R. Bach, Michael I. Jordan\cr
\emph{Predictive low-rank decomposition for kernel methods.}\cr
Proceedings of the Twenty-second International Conference on Machine Learning (ICML) 2005\cr
\url{http://www.di.ens.fr/~fbach/bach_jordan_csi.pdf}
}
\author{Alexandros Karatzoglou (based on Matlab code by
Francis Bach)\cr
\email{alexandros.karatzoglou@ci.tuwien.ac.at}}
\seealso{\code{\link{inchol}}, \code{\link{chol}}, \code{\link{csi-class}}}
\examples{
data(iris)
## create multidimensional y matrix
yind <- t(matrix(1:3,3,150))
ymat <- matrix(0, 150, 3)
ymat[yind==as.integer(iris[,5])] <- 1
datamatrix <- as.matrix(iris[,-5])
# initialize kernel function
rbf <- rbfdot(sigma=0.1)
rbf
Z <- csi(datamatrix,ymat, kernel=rbf, rank = 30)
dim(Z)
pivots(Z)
# calculate kernel matrix
K <- crossprod(t(Z))
# difference between approximated and real kernel matrix
(K - kernelMatrix(kernel=rbf, datamatrix))[6,]
}
\keyword{methods}
\keyword{algebra}
\keyword{array}
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