chol.reduce.rd

这是核学习的一个基础软件包

\name{chol.reduce}
\alias{chol.reduce}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{Incomplete Cholesky decomposition}
\description{
  \code{chol.reduce} computes the incomplete Cholesky decomposition
  of the kernel matrix from a data matrix. 
}
\usage{
chol.reduce(x, kernel="rbfdot", kpar=list(sigma=0.1), tol = 0.001, max.iter = dim(x)[1], verbose = 0)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
  \item{x}{The data matrix indexed by row}
  \item{kernel}{the kernel function used in training and predicting.
    This parameter can be set to any function, of class kernel, which computes a dot product 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)
      \item \code{polydot} (Polynomial kernel function)
      \item \code{vanilladot} (Linear kernel function)
      \item \code{tanhdot} (Hyperbolic tangent kernel function)
    }
    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. For valid parameters for existing kernels are :
    \itemize{
      \item \code{sigma} (inverse kernel width for the Radial Basis kernel function "rbfdot")
      \item \code{degree, scale, offset} (for the Polynomial kernel "polydot")
      \item \code{scale, offset} (for the Hyperbolic tangent kernel
      function "tanhdot")
    }
    Hyper-parameters for user defined kernels can be passed through the
    kpar parameter as well.
  }
  
  \item{tol}{algorithm stops when remaining pivots bring less accuracy
    then \code{tol} (default: 0.001)}
  \item{max.iter}{maximum number of iterations }
  \item{verbose}{print info on algorithm convergence }
}
\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.}
\value{
  An S4 object of class "inc.chol" 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{diag.residues}{Residuals left on the diagonal}
  \item{maxresiduals}{Residuals picked for pivoting}

  slots can be accessed either by \code{object@slot}
or by accessor functions with the same name (e.g. \code{pivots(object))}}
\references{ \item
      Francis R. Bach, Michael I. Jordan\cr
      \emph{Kernel Independent Component Analysis}\cr
      Journal of Machine Learning Research  3, 1-48\cr
      \url{http://www.jmlr.org/papers/volume3/bach02a/bach02a.pdf}
    }
 }
\author{Alexandros Karatzoglou (based on Matlab code by 
  S.V.N. (Vishy) Vishwanathan and Alex Smola)\cr
\email{[e-mail]}}

\seealso{\code{\link{chol}}}
\examples{

data(iris)
datamatrix <- as.matrix(iris[,-5])
# initialize kernel function
rbf <- rbfdot(sigma=0.1)
rbf
Z <- chol.reduce(datamatrix,kernel=rbf)
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{algebra}
\keyword{array}