KhatriRao: Khatri-Rao Matrix Product

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 = "*", make.dimnames = FALSE)

Arguments

X,Y

matrices of with the same number of columns.

FUN

the (name of the) function to be used for the column-wise Kronecker products, see kronecker, defaulting to the usual multiplication.

make.dimnames

logical indicating if the result should inherit dimnames from X and Y in a simple way.

Value

a "CsparseMatrix", say R, the Khatri-Rao product of X ($n x k$) and Y ($m x k$), is of dimension $(n*m) x k$, where the j-th column, R[,j] is the kronecker product kronecker(X[,j], Y[,j]).

References

Khatri, C. G., and Rao, C. Radhakrishna (1968) Solutions to Some Functional Equations and Their Applications to Characterization of Probability Distributions. Sankhya: Indian J. Statistics, Series A 30, 167--180.

Liu, Shuangzhe, and Gõtz Trenkler (2008) Hadamard, Khatri-Rao, Kronecker and Other Matrix Products. International J. Information and Systems Sciences 4, 160--177.

Bader, Brett W, and Tamara G Kolda (2008) Efficient MATLAB Computations with Sparse and Factored Tensors. SIAM J. Scientific Computing 30, 205--231.

Examples

Run this code

## 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,"nMatrix")
nd <- as(d,"nMatrix")
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[] <- 0 # all 0 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)))