#' Minimum Volume Embedding
#'
#' Minimum Volume Embedding (MVE) is a nonlinear dimension reduction
#' algorithm that exploits semidefinite programming (SDP), like MVU/SDE.
#' Whereas MVU aims at stretching through all direction by maximizing
#' \eqn{\sum \lambda_i}, MVE only opts for unrolling the top eigenspectrum
#' and chooses to shrink left-over spectral dimension. For ease of use,
#' unlike kernel PCA, we only made use of Gaussian kernel for MVE.
#'
#' @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 knn size of \eqn{k}-nn neighborhood.
#' @param kwidth bandwidth for Gaussian kernel.
#' @param preprocess an additional option for preprocessing the data.
#' Default is "null". See also \code{\link{aux.preprocess}} for more details.
#' @param tol stopping criterion for incremental change.
#' @param maxiter maximum number of iterations allowed.
#'
#' @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.}
#' }
#'
#' @examples
#' \dontrun{
#' ## use a small subset of iris data
#' set.seed(100)
#' id = sample(1:150, 50)
#' X = as.matrix(iris[id,1:4])
#' lab = as.factor(iris[id,5])
#'
#' ## try different connectivity levels
#' output1 <- do.mve(X, knn=5)
#' output2 <- do.mve(X, knn=10)
#' output3 <- do.mve(X, knn=20)
#'
#' ## Visualize two comparisons
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(output1$Y, main="knn:k=5", pch=19, col=lab)
#' plot(output2$Y, main="knn:k=10", pch=19, col=lab)
#' plot(output3$Y, main="knn:k=20", pch=19, col=lab)
#' par(opar)
#' }
#'
#' @references
#' \insertRef{shaw_minimum_2007}{Rdimtools}
#'
#' @seealso \code{\link{do.mvu}}
#' @author Kisung You
#' @rdname nonlinear_MVE
#' @concept nonlinear_methods
#' @export
do.mve <- function(X, ndim=2, knn=ceiling(nrow(X)/10), kwidth=1.0,
preprocess=c("null","center","scale","cscale","whiten","decorrelate"),
tol=1e-4, maxiter=10){
#------------------------------------------------------------------------
## PARAMETER CHECK
# 1. X : data matrix
aux.typecheck(X)
# 2. ndim : target dimension
if (!check_ndim(ndim,ncol(X))){
stop("* do.mve : 'ndim' is a positive integer in [1,#(covariates)].")
}
ndim = as.integer(ndim)
# 3. knn : the number for k-nn graph
if (length(as.vector(knn))!=1){
stop("* do.mve : knn should be a constant integer number.")
}
knn = as.integer(knn)
if ((knn<1)||(knn>=nrow(X))||(is.na(knn))||(is.infinite(knn))){
stop("* do.mve : knn should be [1,#(covariates)).")
}
# 4. kwidth : kernel bandwidth
if (length(as.vector(kwidth))!=1){
stop("* do.mve : kernel bandwidth should be a constant number.")
}
if ((kwidth < 0)||(is.na(kwidth))||(is.infinite(kwidth))){
stop("* do.mve : kernel bandwidth should be a nonnegative real number.")
}
ktype = c("gaussian",as.double(kwidth))
# 5. tol : tolerance level for stopping criterion
if (length(as.vector(tol))!=1){
stop("* do.mve : 'tol' should be a number.")
}
if ((tol<=0)||(tol>=1)||(is.na(tol))||(is.infinite(tol))){
stop("* do.mve : 'tol' should be a small positive number, possibly in (0,1).")
}
# 6. preprocess
algpreprocess = match.arg(preprocess)
# 7. maxiter
maxiter = round(maxiter)
if (length(as.vector(maxiter))!=1){
stop("* do.mve : 'maxiter' should be a constant integer.")
}
if ((maxiter<3)||(is.na(maxiter))||(is.infinite(maxiter))){
stop("* do.mve : 'maxiter' should be a relatively large positive integer.")
}
#------------------------------------------------------------------------
## PREPROCESSING
# 1. datapreprocessing
tmplist = (X,type=algpreprocess,algtype="nonlinear")
trfinfo = tmplist$info
pX = tmplist$pX
# 2. form affinity matrix A
KernelMap = aux.kernelcov(pX,ktype)
A = KernelMap$K
# 3. use A to find a binary connectivity matrix C via k-nearest neighbors
nbdtype = c("knn",knn)
nbdstruct = aux.graphnbdD(A,type=nbdtype,symmetric="union")
C = nbdstruct$mask # TRUE for connected, FALSE not connected.
#------------------------------------------------------------------------
## MAIN COMPUTATION
# 1. initialize K=A
N = nrow(A)
Kold = A
cvtgap = 1000
itercount = 1
while (cvtgap > tol){
# 2. solve for the eigenvectors of K
eigKold = eigen(Kold)$vectors
B1 = eigKold[,1:ndim]
B2 = eigKold[,(ndim+1):ncol(eigKold)]
B = -(B1%*%t(B1))+(B2%*%t(B2))
# 3. solve SDP via Rcsdp
Knew = mve_single_csdp(A,B,C)
# Knew = mve_single_cvxr(A,B,C)
# 4. update the cvtgap : I will use Frobenius norm
cvtgap = base::norm(Kold-Knew,type="F")
# 5. update Kold -> we will use Kold forever
Kold = Knew
itercount = itercount+1
if (itercount > maxiter){
cvtgap = tol/100;
}
}
# 7. now, Kold is our kernel matrix, so apply Kernel PCA
tY = aux.kernelprojection(Kold, ndim)
#------------------------------------------------------------------------
## RETURN OUTPUTKnew = as.matrix(solprob$getValue(Ktmp), nrow=N)
result = list()
result$Y = t(tY)
result$trfinfo = trfinfo
return(result)
}
#' @keywords internal
#' @noRd
mve_single_cvxr <- function(A, B, C){
N = nrow(B)
Ktmp = CVXR::Variable(N,N,PSD=TRUE)
obj = Maximize(matrix_trace(Ktmp%*%B))
constr1 = list(CVXR::sum_entries(Ktmp)==0)
constr2 = list()
iter = 1
for (i in 1:(N-1)){
for (j in (i+1):N){
if (C[i,j]){ # if two nodes are connected
constr2[[iter]] = ((Ktmp[i,i]+Ktmp[j,j]-Ktmp[i,j]-Ktmp[j,i])==(A[i,i]+A[j,j]-A[i,j]-A[j,i]))
}
iter = iter+1 # update iteration counter
}
}
prob = CVXR::Problem(obj, c(constr1, constr2))
solprob = solve(prob)
Knew = as.matrix(solprob$getValue(Ktmp), nrow=N)
return(Knew)
}
#' @keywords internal
#' @noRd
mve_single_csdp <- function(A, B, C){
# 1. settings
N = nrow(B)
nconstraints = sum(C)/2
# setC = list(B) ::{Rcsdp/ADMM}
setC = -B
setK = list(type="s", size=N)
# 2. iterating for conditions
# 2-1. setup
setA = list()
setb = c()
# 2-2. iterative conditions
iter = 1
for (i in 1:(N-1)){
for (j in (i+1):N){
if (C[i,j]==TRUE){
# tmpA = list(simple_triplet_sym_matrix(i=c(i,i,j,j),j=c(i,j,i,j),v=c(1,-1,-1,1),n=N))
tmpA = alt_triplet_matrix(i=c(i,i,j,j),j=c(i,j,i,j),v=c(1,-1,-1,1),n=N)
tmpb = A[i,i]+A[j,j]-A[i,j]-A[j,i]
setA[[iter]] = tmpA
setb[iter] = tmpb
iter = iter+1
}
}
}
# 2-3. sum to zero
# setA[[iter]] = list(matrix(1,N,N)/N)
setA[[iter]] = matrix(1,N,N)/N
setb[iter] = 0
# outCSDP = (csdp(setC,setA,setb,setK, csdp.control(printlevel=0)))
outCSDP = ADMM::admm.sdp(setC, setA, setb)
return(outCSDP$X)
# return(matrix(outCSDP$X[[1]], nrow=N))
}
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