KhatriRao.Rd
\name{KhatriRao}
\title{Khatri-Rao Matrix Product}
%
\keyword{algebra}
\keyword{arith}
\keyword{array}
\keyword{utilities}
%
\alias{KhatriRao}
%
\description{
Computes Khatri-Rao products for any kind of matrices.
The Khatri-Rao product is a column-wise Kronecker product. Originally
introduced by Khatri and Rao (1968), it has many different applications,
see Liu and Trenkler (2008) for a survey. Notably, it is used in
higher-dimensional tensor decompositions, see Bader and Kolda (2008).
}
\usage{
KhatriRao(X, Y = X, FUN = "*", sparseY = TRUE, make.dimnames = FALSE)
}
\arguments{
\item{X,Y}{matrices of with the same number of columns.}
\item{FUN}{the (name of the) \code{\link{function}} to be used for the
column-wise Kronecker products, see \code{\link{kronecker}},
defaulting to the usual multiplication.}
\item{sparseY}{logical specifying if \code{Y} should be coerced and
treated as \code{\linkS4class{sparseMatrix}}. Set this to
\code{FALSE}, e.g., to distinguish structural zeros from zero entries.}
\item{make.dimnames}{logical indicating if the result should inherit
\code{\link{dimnames}} from \code{X} and \code{Y} in a simple way.}
}
%\details{}
\value{
a \code{"\linkS4class{CsparseMatrix}"}, say \code{R}, the Khatri-Rao
product of \code{X} (\eqn{n \times k}{n x k}) and \code{Y} (\eqn{m
\times k}{m x k}), is of dimension \eqn{(n\cdot m) \times k}{(n*m) x k},
where the j-th column, \code{R[,j]} is the kronecker product
\code{\link{kronecker}(X[,j], Y[,j])}.
}
\note{%% TODO? Could make it generic, and have dense and sparse methods
The current implementation is efficient for large sparse matrices.
}
\references{
Khatri, C. G., and Rao, C. Radhakrishna (1968)
Solutions to Some Functional Equations and Their Applications to
Characterization of Probability Distributions.
\emph{Sankhya: Indian J. Statistics, Series A} \bold{30}, 167--180.
Bader, Brett W, and Tamara G Kolda (2008)
Efficient MATLAB Computations with Sparse and Factored Tensors.
\emph{SIAM J. Scientific Computing} \bold{30}, 205--231.
}
\author{
Original by Michael Cysouw, Univ. Marburg;
minor tweaks, bug fixes etc, by Martin Maechler.
}
\seealso{
\code{\link{kronecker}}.
}
\examples{
## Example with very small matrices:
m <- matrix(1:12,3,4)
d <- diag(1:4)
KhatriRao(m,d)
KhatriRao(d,m)
dimnames(m) <- list(LETTERS[1:3], letters[1:4])
KhatriRao(m,d, make.dimnames=TRUE)
KhatriRao(d,m, make.dimnames=TRUE)
dimnames(d) <- list(NULL, paste0("D", 1:4))
KhatriRao(m,d, make.dimnames=TRUE)
KhatriRao(d,m, make.dimnames=TRUE)
dimnames(d) <- list(paste0("d", 10*1:4), paste0("D", 1:4))
(Kmd <- KhatriRao(m,d, make.dimnames=TRUE))
(Kdm <- KhatriRao(d,m, make.dimnames=TRUE))
nm <- as(m, "nsparseMatrix")
nd <- as(d, "nsparseMatrix")
KhatriRao(nm,nd, make.dimnames=TRUE)
KhatriRao(nd,nm, make.dimnames=TRUE)
stopifnot(dim(KhatriRao(m,d)) == c(nrow(m)*nrow(d), ncol(d)))
## border cases / checks:
zm <- nm; zm[] <- FALSE # all FALSE matrix
stopifnot(all(K1 <- KhatriRao(nd, zm) == 0), identical(dim(K1), c(12L, 4L)),
all(K2 <- KhatriRao(zm, nd) == 0), identical(dim(K2), c(12L, 4L)))
d0 <- d; d0[] <- 0; m0 <- Matrix(d0[-1,])
stopifnot(all(K3 <- KhatriRao(d0, m) == 0), identical(dim(K3), dim(Kdm)),
all(K4 <- KhatriRao(m, d0) == 0), identical(dim(K4), dim(Kmd)),
all(KhatriRao(d0, d0) == 0), all(KhatriRao(m0, d0) == 0),
all(KhatriRao(d0, m0) == 0), all(KhatriRao(m0, m0) == 0),
identical(dimnames(KhatriRao(m, d0, make.dimnames=TRUE)), dimnames(Kmd)))
## a matrix with "structural" and non-structural zeros:
m01 <- new("dgCMatrix", i = c(0L, 2L, 0L, 1L), p = c(0L, 0L, 0L, 2L, 4L),
Dim = 3:4, x = c(1, 0, 1, 0))
D4 <- Diagonal(4, x=1:4) # "as" d
DU <- Diagonal(4)# unit-diagonal: uplo="U"
(K5 <- KhatriRao( d, m01))
K5d <- KhatriRao( d, m01, sparseY=FALSE)
K5Dd <- KhatriRao(D4, m01, sparseY=FALSE)
K5Ud <- KhatriRao(DU, m01, sparseY=FALSE)
(K6 <- KhatriRao(diag(3), t(m01)))
K6D <- KhatriRao(Diagonal(3), t(m01))
K6d <- KhatriRao(diag(3), t(m01), sparseY=FALSE)
K6Dd <- KhatriRao(Diagonal(3), t(m01), sparseY=FALSE)
stopifnot(exprs = {
all(K5 == K5d)
identical(cbind(c(7L, 10L), c(3L, 4L)),
which(K5 != 0, arr.ind = TRUE, useNames=FALSE))
identical(K5d, K5Dd)
identical(K6, K6D)
all(K6 == K6d)
identical(cbind(3:4, 1L),
which(K6 != 0, arr.ind = TRUE, useNames=FALSE))
identical(K6d, K6Dd)
})
}