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#' Complete Neighborhood Preserving Embedding
#'
#' One of drawbacks of Neighborhood Preserving Embedding (NPE) is the small-sample-size problem
#' under high-dimensionality of original data, where singular matrices to be decomposed suffer from
#' rank deficiency. Instead of applying PCA as a preprocessing step, Complete NPE (CNPE) transforms the
#' singular generalized eigensystem computation of NPE into two eigenvalue decomposition problems.
#'
#' @param X an \eqn{(n\times p)} matrix or data frame whose rows are observations
#' and columns represent independent variables.
#' @param ndim an integer-valued target dimension.
#' @param type a vector of neighborhood graph construction. Following types are supported;
#' \code{c("knn",k)}, \code{c("enn",radius)}, and \code{c("proportion",ratio)}.
#' Default is \code{c("proportion",0.1)}, connecting about 1/10 of nearest data points
#' among all data points. See also \code{\link{aux.graphnbd}} for more details.
#' @param preprocess an additional option for preprocessing the data.
#' Default is "center". See also \code{\link{aux.preprocess}} for more details.
#'
#' @return a named list containing
#' \describe{
#' \item{Y}{an \eqn{(n\times ndim)} matrix whose rows are embedded observations.}
#' \item{trfinfo}{a list containing information for out-of-sample prediction.}
#' \item{projection}{a \eqn{(p\times ndim)} whose columns are basis for projection.}
#' }
#'
#' @examples
#' \donttest{
#' ## generate data of 3 types with clear difference
#' dt1 = aux.gensamples(n=20)-50
#' dt2 = aux.gensamples(n=20)
#' dt3 = aux.gensamples(n=20)+50
#' lab = rep(1:3, each=20)
#'
#' ## merge the data
#' X = rbind(dt1,dt2,dt3)
#'
#' ## try different numbers for neighborhood size
#' out1 = do.cnpe(X, type=c("proportion",0.10))
#' out2 = do.cnpe(X, type=c("proportion",0.25))
#' out3 = do.cnpe(X, type=c("proportion",0.50))
#'
#' ## visualize
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(out1$Y, col=lab, pch=19, main="CNPE::10% connected")
#' plot(out2$Y, col=lab, pch=19, main="CNPE::25% connected")
#' plot(out3$Y, col=lab, pch=19, main="CNPE::50% connected")
#' par(opar)
#' }
#'
#' @references
#' \insertRef{wang_complete_2010}{Rdimtools}
#'
#' @rdname linear_CNPE
#' @author Kisung You
#' @concept linear_methods
#' @export
do.cnpe <- function(X, ndim=2, type=c("proportion",0.1), preprocess=c("center","scale","cscale","decorrelate","whiten")){
#------------------------------------------------------------------------
## PREPROCESSING
# 1. data matrix
aux.typecheck(X)
n = nrow(X)
p = ncol(X)
# 2. ndim
ndim = as.integer(ndim)
if (!check_ndim(ndim,p)){
stop("* do.cnpe : 'ndim' is a positive integer in [1,#(covariates)].")
}
# 3. type
nbdtype = type
nbdsymmetric = "union"
# 4. preprocess
if (missing(preprocess)){
algpreprocess = "center"
} else {
algpreprocess = match.arg(preprocess)
}
#------------------------------------------------------------------------
## COMPUTATION : PRELIMINARY and LLE step
# 1. preprocessing of data : note that output pX still has (n-by-p) format
tmplist = (X,type=algpreprocess,algtype="linear")
trfinfo = tmplist$info
pX = tmplist$pX
# 2. neighborhood information
nbdstruct = aux.graphnbd(pX,method="euclidean",
type=nbdtype,symmetric=nbdsymmetric)
# 3. LLE computation
regparam = 1.0
W = array(0,c(n,n))
for (i in 1:n){
# 3-1. separate target mask vector
tgtidx = which(nbdstruct$mask[i,])
# 3-2. select data
# For convenience, target matrix is transposed for Armadillo
vec_tgt = pX[i,]
mat_tgt = t(pX[tgtidx,])
k = ncol(mat_tgt)
# 3-3. no automatic regularization
W[i,tgtidx] = method_lleW(mat_tgt,vec_tgt,regparam);
}
# 4. preliminary rank determination
diagN = diag(n)
M = t(diagN-W)%*%(diagN-W)
St = (t(pX)%*%M%*%pX) + (t(pX)%*%pX)
r = round(aux_rank(St)) # as.integer(Matrix::rankMatrix (St))
if (r < ndim){
message("* do.cnpe : intrinsic rank of matrix St is smaller than 'ndim'.")
ndim = r
}
#------------------------------------------------------------------------
## COMPUTATION : MAIN COMPUTATION FOR CNPE
# 1. EVD for t(Xtilde)%*%Xtilde
# select Vr and vecSig1
Xtilde = t(pX)%*%cbind(t(diagN-W),diagN) # (D x 2N)
Xcost = t(Xtilde)%*%Xtilde # (2N x 2N)
eigXcost = base::eigen(Xcost)
Vr = eigXcost$vectors[,1:r]
vecSig1 = as.vector(eigXcost$values[1:r])
# 2. compute Ur and Sctilde
invSig1half = diag(1/sqrt(vecSig1))
Ur = Xtilde%*%Vr%*%invSig1half
Sctilde = invSig1half%*%t(Ur)%*%t(pX)%*%pX%*%Ur%*%invSig1half
# 3. decompose Sctilde and denote it as Wmat
Wmat = base::eigen(Sctilde)$vectors
# 4. use first ndim unitary vectors
resmat = Ur%*%invSig1half%*%Wmat
projection = aux.adjprojection(resmat[,1:ndim])
#------------------------------------------------------------------------
## RETURN
result = list()
result$Y = pX%*%projection
result$trfinfo = trfinfo
result$projection = projection
return(result)
}
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