tensor_predictors/tensorPredictors/R/dist_subspace.R

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#' Subspace distance
#'
#' @param A,B Basis matrices as representations of elements of the Grassmann
#' manifold.
#' @param is.ortho Boolean to specify if \eqn{A} and \eqn{B} are semi-orthogonal.
#' If false, the projection matrices are computed as
#' \deqn{P_A = A (A' A)^{-1} A'}
#' otherwise just \eqn{P_A = A A'} since \eqn{A' A} is the identity.
#' @param normalize Boolean to specify if the distance shall be normalized.
#' Meaning, the maximal distance scaled to be \eqn{1} independent of dimensions.
#'
#' @seealso
#' K. Ye and L.-H. Lim (2016) "Schubert varieties and distances between
#' subspaces of different dimensions" <arXiv:1407.0900>
#'
#' @export
dist.subspace <- function (A, B, is.ortho = FALSE, normalize = FALSE,
tol = sqrt(.Machine$double.eps)
) {
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if (!is.matrix(A)) A <- as.matrix(A)
if (!is.matrix(B)) B <- as.matrix(B)
if (!is.ortho) {
qrA <- qr(A, tol)
if (qrA$rank < ncol(A)) {
A <- qr.Q(qrA)[, abs(diag(qr.R(qrA))) > tol, drop = FALSE]
} else {
A <- qr.Q(qrA)
}
qrB <- qr(B, tol)
if (qrB$rank < ncol(B)) {
B <- qr.Q(qrB)[, abs(diag(qr.R(qrB))) > tol, drop = FALSE]
} else {
B <- qr.Q(qrB)
}
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}
PA <- tcrossprod(A, A)
PB <- tcrossprod(B, B)
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if (normalize) {
rankSum <- ncol(A) + ncol(B)
c <- 1 / sqrt(min(rankSum, 2 * nrow(A) - rankSum))
} else {
c <- sqrt(2)
}
c * norm(PA - PB, type = "F")
}