R/nonlinear_MVU.R

In Rdimtools: Dimension Reduction and Estimation Methods

#' Maximum Variance Unfolding / Semidefinite Embedding
#'
#' The method of Maximum Variance Unfolding(MVU), also known as Semidefinite Embedding(SDE) is, as its names suggest,
#' to exploit semidefinite programming in performing nonlinear dimensionality reduction by \emph{unfolding}
#' neighborhood graph constructed in the original high-dimensional space. Its unfolding generates a gram
#' matrix \eqn{K} in that we can choose from either directly finding embeddings (\code{"spectral"}) or
#' use again Kernel PCA technique (\code{"kpca"}) to find low-dimensional representations.
#'
#' @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 "null". See also \code{\link{aux.preprocess}} for more details.
#' @param projtype type of method for projection; either \code{"spectral"} or \code{"kpca"} used.
#'
#' @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
#' \donttest{
#' ## 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.mvu(X, type=c("proportion", 0.10))
#' output2 <- do.mvu(X, type=c("proportion", 0.25))
#' output3 <- do.mvu(X, type=c("proportion", 0.50))
#'
#' ## visualize three different projections
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(output1$Y, main="10% connected", pch=19, col=lab)
#' plot(output2$Y, main="25% connected", pch=19, col=lab)
#' plot(output3$Y, main="50% connected", pch=19, col=lab)
#' par(opar)
#' }
#'
#' @references
#' \insertRef{weinberger_unsupervised_2006}{Rdimtools}
#'
#' @author Kisung You
#' @aliases do.sde
#' @rdname nonlinear_MVU
#' @concept nonlinear_methods
#' @export
do.mvu <- function(X,ndim=2,type=c("proportion",0.1),
                   preprocess=c("null","center","scale","cscale","decorrelate","whiten"),
                   projtype=c("spectral","kpca")){
  # 1. typecheck is always first step to perform.
  aux.typecheck(X)
  if ((!is.numeric(ndim))||(ndim<1)||(ndim>ncol(X))||is.infinite(ndim)||is.na(ndim)){
    stop("* do.mvu : 'ndim' is a positive integer in [1,#(covariates)].")
  }
  ndim = as.integer(ndim)

  # extra
  if (missing(projtype)){
    projtype = "spectral"
  } else {
    projtype = match.arg(projtype)
  }

  # 2. ... parameters
  # 2-1. aux.graphnbd
  #   type : vector of c("knn",k), c("enn",radius), or c("proportion",ratio)
  # 2-2. mvu itself
  #   preprocess : 'center','decorrelate', 'null', or 'whiten'
  nbdtype = type
  nbdsymmetric = "union"

  if (missing(preprocess)){
    algpreprocess = "center"
  } else {
    algpreprocess = match.arg(preprocess)
  }

  # 3. process : data preprocessing
  tmplist = aux.preprocess.hidden(X,type=algpreprocess,algtype="nonlinear")
  trfinfo = tmplist$info
  pX      = tmplist$pX
  n = nrow(pX)
  p = ncol(pX)

  # 4. process : neighborhood selection + ALPHA for SDE/MVU
  #            : make a triangle.
  nbdstruct = aux.graphnbd(pX,method="euclidean",
                           type=nbdtype,symmetric="union")
  M1 = nbdstruct$mask
  M2 = M1
  for (it in 1:nrow(M1)){
    for (i in 1:ncol(M1)){
      for (j in 1:ncol(M1)){
        if ((M1[it,i]==TRUE)&&(M1[it,j]==TRUE)){
          M2[i,j] = TRUE
          M2[j,i] = TRUE
        }
      }
    }
  }
  diag(M2) = FALSE

  # 5. Compute K
  #   5-1. basic settings
  N  = ncol(M2) # number of data points
  nconstraints = sum(M2)/2
  D2 = ((as.matrix(dist(pX)))^2)
  # C  = list(.simple_triplet_diag_sym_matrix(1,N))
  C  = -diag(N) # ADMM minimization/Rcsdp maximization
  K  = list(type="s",size=N)
  #   5-2. iterative computation
  iter = 1
  A = list()
  b = c()
  for (it1 in 1:nrow(M2)){
    for (it2 in it1:ncol(M2)){
      if (M2[it1,it2]==TRUE){
        # tmpA = list(simple_triplet_sym_matrix(i=c(it1,it1,it2,it2),j=c(it1,it2,it1,it2),v=c(1,-1,-1,1),n=N))
        tmpA = alt_triplet_matrix(i=c(it1,it1,it2,it2),j=c(it1,it2,it1,it2),v=c(1,-1,-1,1),n=N)
        tmpb = D2[it1,it2]
        A[[iter]] = tmpA
        b[iter]   = tmpb
        iter = iter+1
      }
    }
  }
  #   5-3. final update for mean centered constraint
  A[[iter]] = matrix(1,N,N)/N
  b[iter]   = 0
  # outCSDP = csdp(C,A,b,K,csdp.control(printlevel=0))
  outCSDP = ADMM::admm.sdp(C,A,b)
  # KK      = as.matrix(outCSDP$X[[1]])
  KK      = outCSDP$X

  if (all(projtype=="spectral")){
    # 6. Embedding : Spectral Method, directly from K
    KKeigen = eigen(KK)
    eigvals = KKeigen$values
    eigvecs = KKeigen$vectors

    tY = (diag(sqrt(eigvals[1:ndim])) %*% t(eigvecs[,1:ndim]))
  } else if (all(projtype=="kpca")){
    # 7. Embedding : Kernel PCA method
    tY = aux.kernelprojection(KK, ndim)
  } else {
    stop("* do.mvu : 'projtype' should be either 'spectral' or 'kpca'.")
  }

  # 8. Return output
  result = list()
  result$Y = t(tY)
  result$trfinfo = trfinfo
  return(result)
}

# triplet alternative -----------------------------------------------------
#' @keywords internal
#' @noRd
alt_triplet_matrix <- function(i,j,v,n){
  output = array(0,c(n,n))
  nobj = length(i)
  for (it in 1:nobj){
    output[i[it],j[it]] = v[it]
  }
  return(output)
}