R/scPCA.R

In scPCA: Sparse Contrastive Principal Component Analysis

#' Sparse Contrastive Principal Component Analysis
#'
#' @description Given target and background data frames or matrices,
#'  \code{scPCA} will perform the sparse contrastive principal component
#'  analysis (scPCA) of the target data for a given number of eigenvectors, a
#'  vector of real-valued contrast parameters, and a vector of sparsity inducing
#'  penalty terms.
#'  
#'  If instead you wish to perform contrastive principal component analysis
#'  (cPCA), set the \code{penalties} argument to \code{0}. So long as the
#'  \code{n_centers} parameter is larger than one, the automated hyperparameter
#'  tuning heuristic described in \insertCite{boileau2020;textual}{scPCA} is
#'  used. Otherwise, the semi-automated approach of
#'  \insertCite{abid2018exploring;textual}{scPCA} is used to select the
#'  appropriate hyperparameter.
#'
#' @param target The target (experimental) data set, in a standard format such
#'  as a \code{data.frame} or \code{matrix}. \code{dgCMatrix} and
#'  \code{DelayedMatrix} objects are also supported.
#' @param background The background data set, in a standard format such as a
#'  \code{data.frame} or \code{matrix}. The features must match the features of
#'  the target data set. \code{dgCMatrix} and \code{DelayedMatrix} objects are
#'  also supported.
#' @param center A \code{logical} indicating whether the target and background
#'  data sets' features should be centered to mean zero.
#' @param scale A \code{logical} indicating whether the target and background
#'  data sets' features should be scaled to unit variance.
#' @param n_eigen A \code{numeric} indicating the number of eigenvectors (or
#'  (sparse) contrastive components) to be computed. Two eigenvectors are
#'  computed by default.
#' @param cv A \code{numeric} indicating the number of cross-validation folds
#'  to use in choosing the optimal contrastive and penalization parameters from
#'  over the grids of \code{contrasts} and \code{penalties}. Cross-validation
#'  is expected to improve the robustness and generalization of the choice of
#'  these parameters. However, it increases the time the procedure costs.
#'  The default is therefore \code{NULL}, corresponding to no cross-validation.
#' @param alg A \code{character} indicating the sparse PCA algorithm used to
#'  sparsify the contrastive loadings. Currently supports \code{iterative} for
#'  the \insertCite{zou2006sparse;textual}{scPCA} implementation, \code{var_proj}
#'  for the non-randomized \insertCite{erichson2018sparse;textual}{scPCA}
#'  solution, and \code{rand_var_proj} for the randomized
#'  \insertCite{erichson2018sparse;textual}{scPCA} implementation. Defaults to
#'  \code{iterative}.
#' @param contrasts A \code{numeric} vector of the contrastive parameters. Each
#'  element must be a unique, non-negative real number. By default, 40
#'  logarithmically spaced values between 0.1 and 1000 are used. If a single
#'  value is provided and \code{penalties} is set to 0, then \code{n_centers},
#'  \code{clust_method}, \code{max_iter}, \code{linkage_method},
#'  \code{n_medoids}, and \code{parallel} can be safely ignored.
#' @param penalties A \code{numeric} vector of the L1 penalty terms on the
#'  loadings. The default is to use 20 equidistant values between 0.05 and 1.
#'  If \code{penalties} is set to 0, then cPCA is performed in place of scPCA.
#'  See \code{contrasts} and \code{n_centers} arguments for more infotmation.
#' @param clust_method A \code{character} specifying the clustering method to
#'  use for choosing the optimal contrastive parameter. Currently, this is
#'  limited to either k-means, partitioning around medoids (PAM), and
#'  hierarchical clustering. The default is k-means clustering.
#' @param n_centers A \code{numeric} giving the number of centers to use in the
#'  clustering algorithm. If set to 1, cPCA, as first proposed by
#'  \insertCite{erichson2018sparse;textual}{scPCA}, is performed, regardless of
#'  what the \code{penalties} argument is set to.
#' @param max_iter A \code{numeric} giving the maximum number of iterations to
#'   be used in k-means clustering. Defaults to 10.
#' @param linkage_method A \code{character} specifying the agglomerative
#'   linkage method to be used if \code{clust_method = "hclust"}. The options
#'   are \code{ward.D2}, \code{single}, \code{complete}, \code{average},
#'   \code{mcquitty}, \code{median}, and \code{centroid}. The default is
#'   \code{complete}.
#' @param n_medoids A \code{numeric} indicating the number of medoids to
#'  consider if \code{n_centers} is set to 1 and \code{contrasts} is a vector of
#'  length 2 or more. The default is 8 medoids.
#' @param parallel A \code{logical} indicating whether to invoke parallel
#'  processing via the \pkg{BiocParallel} infrastructure. The default is
#'  \code{FALSE} for sequential evaluation.
#' @param clusters A \code{numeric} vector of cluster labels for observations in
#'  the \code{target} data. Defaults to \code{NULL}, but is otherwise used to
#'  identify the optimal set of hyperparameters when fitting the scPCA and the
#'  automated version of cPCA. If a vector is provided, the
#'  \code{n_centers}, \code{clust_method}, \code{max_iter},
#'  \code{linkage_method}, and \code{n_medoids} arguments can be safely ignored.
#' @param eigdecomp_tol A \code{numeric} providing the level of precision used by
#'  eigendecompositon calculations. Defaults to \code{1e-10}.
#' @param eigdecomp_iter A \code{numeric} indicating the maximum number of
#'  interations performed by eigendecompositon calculations. Defaults to
#'  \code{1000}.
#'
#' @return A list containing the following components:
#'   \itemize{
#'     \item \code{rotation}: The matrix of variable loadings if \code{n_centers}
#'       is larger than one. Otherwise, a list of rotation matrices is returned,
#'       one for each medoid. The number of medoids is specified by
#'       \code{n_medoids}.
#'     \item \code{x}: The rotated data, centred and scaled if requested,
#'       multiplied by the rotation matrix if \code{n_centers} is larger than
#'       one. Otherwise, a list of rotated data matrices is returned, one for
#'       each medoid. The number of medoids is specified by \code{n_medoids}.
#'     \item contrast: The optimal contrastive parameter.
#'     \item penalty: The optimal L1 penalty term.
#'     \item center: A logical indicating whether the target dataset was centered.
#'     \item scale: A logical indicating whether the target dataset was scaled.
#'   }
#'
#' @importFrom Rdpack reprompt
#' @importFrom origami cross_validate make_folds
#' @importFrom dplyr "%>%" full_join
#' @importFrom purrr reduce
#' @importFrom stringr str_detect
#' @importFrom tibble as_tibble
#'
#' @references
#'   \insertAllCited{}
#'   
#' @export
#'
#' @examples
#' # perform cPCA on the simulated data set
#' scPCA(
#'   target = toy_df[, 1:30],
#'   background = background_df,
#'   contrasts = exp(seq(log(0.1), log(100), length.out = 5)),
#'   penalties = 0,
#'   n_centers = 4
#' )
#'
#' # perform scPCA on the simulated data set
#' scPCA(
#'   target = toy_df[, 1:30],
#'   background = background_df,
#'   contrasts = exp(seq(log(0.1), log(100), length.out = 5)),
#'   penalties = seq(0.1, 1, length.out = 3),
#'   n_centers = 4
#' )
#'
#' # perform cPCA on the simulated data set with known clusters
#' scPCA(
#'   target = toy_df[, 1:30],
#'   background = background_df,
#'   contrasts = exp(seq(log(0.1), log(100), length.out = 5)),
#'   penalties = 0,
#'   clusters = toy_df[, 31]
#' )
#'
#' # cPCA as implemented in Abid et al.
#' scPCA(
#'   target = toy_df[, 1:30],
#'   background = background_df,
#'   contrasts = exp(seq(log(0.1), log(100), length.out = 10)),
#'   penalties = 0,
#'   n_centers = 1
#' )
scPCA <- function(target, background, center = TRUE, scale = FALSE,
                  n_eigen = 2, cv = NULL,
                  alg = c("iterative", "var_proj", "rand_var_proj"),
                  contrasts = exp(seq(log(0.1), log(1000), length.out = 40)),
                  penalties = seq(0.05, 1, length.out = 20),
                  clust_method = c("kmeans", "pam", "hclust"),
                  n_centers = NULL, max_iter = 10, linkage_method = "complete",
                  n_medoids = 8, parallel = FALSE, clusters = NULL,
                  eigdecomp_tol = 1e-10, eigdecomp_iter = 1000) {
  # set defaults
  clust_method <- match.arg(clust_method)
  alg <- match.arg(alg)

  # check arguments to function
  checkArgs(
    target, background, center, scale, n_eigen,
    contrasts, penalties, clust_method, linkage_method,
    clusters, eigdecomp_tol, eigdecomp_iter, n_centers
  )

  if (!is.null(clusters)) {
    # set a dummy value for clusters when cluster labels are passed in
    n_centers <- 2
    
    # coerce clusters argument to an integer
    if (is.factor(clusters)) {
      clusters <- as.numeric(clusters)
    } else if (is.character(clusters)) {
      clusters <- as.numeric(as.factor(clusters))
    }
  }

  if (is.null(cv)) {
    opt_params <- selectParams(
      target = target,
      background = background,
      center = center,
      scale = scale,
      n_eigen = n_eigen,
      alg = alg,
      contrasts = contrasts,
      penalties = penalties,
      clust_method = clust_method,
      n_centers = n_centers,
      max_iter = max_iter,
      linkage_method = linkage_method,
      n_medoids = n_medoids,
      parallel = parallel,
      clusters = clusters,
      eigdecomp_tol = eigdecomp_tol,
      eigdecomp_iter = eigdecomp_iter
    )
    if (length(contrasts) == 1 && length(penalties) == 1 &&
        penalties[[1]] == 0) {
      opt_params <- list(
        rotation = opt_params$rotation,
        x = opt_params$x,
        contrast = opt_params$contrast,
        penalty = opt_params$penalty
      )
    } else if (length(contrasts) == 1 && length(penalties) == 1 &&
               penalties[[1]] != 0) {
      opt_params <- list(
        rotation = opt_params$rotation,
        x = opt_params$x,
        contrast = opt_params$contrast,
        penalty = opt_params$penalty
      )
    } else if (n_centers > 1) {
      max_idx <- which.max(opt_params$ave_sil_widths)
      opt_params <- list(
        rotation = opt_params$rotation[[max_idx]],
        x = opt_params$x[[max_idx]],
        contrast = opt_params$contrast[max_idx],
        penalty = opt_params$penalty[max_idx]
      )
    }
  } else {
    # partition target and background data sets into CV-splits
    folds_target <- origami::make_folds(target, V = cv)
    folds_background <- origami::make_folds(background, V = cv)
    folds <- lapply(seq_len(cv), function(cv_fold) {
      out <- list(
        target = folds_target[[cv_fold]],
        background = folds_background[[cv_fold]]
      )
      return(out)
    })

    # select tuning parameters via cross-validation
    cv_opt_params <- origami::cross_validate(
      cv_fun = cvSelectParams,
      folds = folds,
      target = target,
      background = background,
      center = center,
      scale = scale,
      n_eigen = n_eigen,
      alg = alg,
      contrasts = contrasts,
      penalties = penalties,
      clust_method = clust_method,
      n_centers = n_centers,
      max_iter = max_iter,
      linkage_method = linkage_method,
      n_medoids = n_medoids,
      parallel = parallel,
      clusters = clusters,
      eigdecomp_tol = eigdecomp_tol,
      eigdecomp_iter = eigdecomp_iter,
      use_future = FALSE,
      .combine = FALSE
    )

    # match up tables of contrast-penalty-silhouette across folds and merge
    cv_ave_sil_pairs <- lapply(seq_len(cv), function(v) {
      ave_sil_pairings <- tibble::as_tibble(
        list(
          contrast = cv_opt_params$contrast[[v]],
          penalty = cv_opt_params$penalty[[v]],
          ave_sil_widths = cv_opt_params$ave_sil_widths[[v]]
        )
      )
    }) %>%
      purrr::reduce(dplyr::full_join, by = c("contrast", "penalty"))

    # compute CV-average silhouette width, find maximizer, and select the
    # optimal set of contrastive and penalization parameters
    ave_sil_col_idx <- stringr::str_detect(
      colnames(cv_ave_sil_pairs),
      "ave_sil_width"
    )
    cv_sil_max_idx <- which.max(rowMeans(cv_ave_sil_pairs[, ave_sil_col_idx]))
    cv_opt_params <- cv_ave_sil_pairs[cv_sil_max_idx, c("contrast", "penalty")]

    # re-fit on full data to get more stable estimates
    fit_cv_opt_params <- selectParams(
      target = target,
      background = background,
      center = center,
      scale = scale,
      n_eigen = n_eigen,
      alg = alg,
      contrasts = cv_opt_params$contrast,
      penalties = cv_opt_params$penalty,
      clust_method = clust_method,
      n_centers = n_centers,
      max_iter = max_iter,
      linkage_method = linkage_method,
      n_medoids = n_medoids,
      parallel = parallel,
      clusters = clusters,
      eigdecomp_tol = eigdecomp_tol,
      eigdecomp_iter = eigdecomp_iter
    )
    if (n_centers > 1 && length(penalties) == 1 && penalties[1] == 0) {
      opt_params <- list(
        rotation = fit_cv_opt_params$rotation,
        x = fit_cv_opt_params$x,
        contrast = fit_cv_opt_params$contrast,
        penalty = fit_cv_opt_params$penalty
      )
    } else if (n_centers > 1) {
      opt_params <- list(
        rotation = fit_cv_opt_params$rotation[[1]],
        x = fit_cv_opt_params$x[[1]],
        contrast = fit_cv_opt_params$contrast,
        penalty = fit_cv_opt_params$penalty
      )
    }
  }

  # create output object
  scpca <- list(
    rotation = opt_params$rotation,
    x = opt_params$x,
    contrast = opt_params$contrast,
    penalty = opt_params$penalty,
    center = center,
    scale = scale
  )
  class(scpca) <- "scpca"
  return(scpca)
}

###############################################################################

#' Selection of Contrastive and Penalization Parameters
#'
#' @description A wrapper function for fitting various internal functions to
#'  select the optimal setting of the contrastive and penalization parameters.
#'  For internal use only.
#'
#' @param target The target (experimental) data set, in a standard format such
#'  as a \code{data.frame} or \code{matrix}.
#' @param background The background data set, in a standard format such as a
#'  \code{data.frame} or \code{matrix}. Note that the number of features must
#'  match the number of features in the target data.
#' @param center A \code{logical} indicating whether the target and background
#'  data sets should be centered to mean zero.
#' @param scale A \code{logical} indicating whether the target and background
#'  data sets should be scaled to unit variance.
#' @param n_eigen A \code{numeric} indicating the number of eigenvectors (or
#'  sparse contrastive components) to be computed. The default is to compute
#'  two such eigenvectors.
#' @param alg A \code{character} indicating the SPCA algorithm used to sparsify
#'  the contrastive loadings. Currently supports \code{iterative} for the
#'  \insertCite{zou2006sparse;textual}{scPCA} implementation, \code{var_proj}
#'  for the non-randomized \insertCite{erichson2018sparse;textual}{scPCA}
#'  solution, and \code{rand_var_proj} for the randomized
#'  \insertCite{erichson2018sparse;textual}{scPCA} result.
#' @param contrasts A \code{numeric} vector of the contrastive parameters. Each
#'  element must be a unique non-negative real number. The default is to use 40
#'  logarithmically spaced values between 0.1 and 1000.
#' @param penalties A \code{numeric} vector of the L1 penalty terms on the
#'  loadings. The default is to use 20 equidistant values between 0.05 and 1.
#' @param clust_method A \code{character} specifying the clustering method to
#'  use for choosing the optimal contrastive parameter. Currently, this is
#'  limited to either k-means, partitioning around medoids (PAM), and
#'  hierarchical clustering. The default is k-means clustering.
#' @param n_centers A \code{numeric} giving the number of centers to use in the
#'  clustering algorithm. If set to 1, cPCA, as first proposed by Abid et al.,
#'  is performed, regardless of what the \code{penalties} argument is set to.
#' @param max_iter A \code{numeric} giving the maximum number of iterations to
#'   be used in k-means clustering, defaulting to 10.
#' @param linkage_method A \code{character} specifying the agglomerative
#'   linkage method to be used if \code{clust_method = "hclust"}. The options
#'   are \code{ward.D2}, \code{single}, \code{complete}, \code{average},
#'   \code{mcquitty}, \code{median}, and \code{centroid}. The default is
#'   \code{complete}.
#' @param n_medoids A \code{numeric} indicating the number of medoids to
#'  consider if \code{n_centers} is set to 1. The default is 8 such medoids.
#' @param parallel A \code{logical} indicating whether to invoke parallel
#'  processing via the \pkg{BiocParallel} infrastructure. The default is
#'  \code{FALSE} for sequential evaluation.
#' @param clusters A \code{numeric} vector of cluster labels for observations in
#'  the \code{target} data. Defaults to \code{NULL}, but is otherwise used to
#'  identify the optimal set of hyperparameters when fitting the scPCA and the
#'  automated version of cPCA.
#' @param eigdecomp_tol A \code{numeric} providing the level of precision used by
#'  eigendecompositon calculations. Defaults to \code{1e-10}.
#' @param eigdecomp_iter A \code{numeric} indicating the maximum number of
#'  interations performed by eigendecompositon calculations. Defaults to
#'  \code{1000}.
#'
#' @return Output structure matching either that of \code{\link{fitCPCA}} or
#'  \code{\link{fitGrid}} (or their parallelized variants, namely either
#'  \code{\link{bpFitCPCA}} and \code{link{bpFitGrid}}, respectively).
#'
#' @references
#'   \insertAllCited{}
#'   
#' @keywords internal
selectParams <- function(target, background, center, scale, n_eigen, alg,
                         contrasts, penalties, clust_method, n_centers,
                         max_iter, linkage_method, n_medoids, parallel,
                         clusters, eigdecomp_tol, eigdecomp_iter) {
  
  # call parallelized function variants if so requested
  if (!parallel || (length(penalties) == 1 && length(contrasts) == 1)) {
    # create contrastive covariance matrices
    c_contrasts <- contrastiveCov(
      target = target, background = background, contrasts = contrasts,
      center = center, scale = scale
    )
    if (length(penalties) == 1 && penalties[1] != 0 && length(contrasts) == 1) {
      opt_params <- fitGrid(
        target = target, center = center, scale = scale, alg = alg,
        c_contrasts = c_contrasts, contrasts = contrasts,
        penalties = penalties, n_eigen = n_eigen,
        clust_method = clust_method, n_centers = n_centers,
        max_iter = max_iter, linkage_method = linkage_method,
        clusters = clusters, eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter = eigdecomp_iter
      )
    } else if (n_centers == 1 || (length(penalties) == 1 && penalties[1] == 0
         && length(contrasts) == 1)) {
      opt_params <- fitCPCA(
        target = target, center = center, scale = scale,
        c_contrasts = c_contrasts, contrasts = contrasts,
        n_eigen = n_eigen, n_medoids = n_medoids,
        eigdecomp_tol = eigdecomp_tol, eigdecomp_iter = eigdecomp_iter
      )
    } else {
      opt_params <- fitGrid(
        target = target, center = center, scale = scale, alg = alg,
        c_contrasts = c_contrasts, contrasts = contrasts,
        penalties = penalties, n_eigen = n_eigen,
        clust_method = clust_method, n_centers = n_centers,
        max_iter = max_iter, linkage_method = linkage_method,
        clusters = clusters, eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter = eigdecomp_iter
      )
    }
  } else {
    # create contrastive covariance matrices
    c_contrasts <- bpContrastiveCov(
      target = target, background = background, contrasts = contrasts,
      center = center, scale = scale
    )
    if (n_centers == 1) {
      opt_params <- bpFitCPCA(
        target = target, center = center, scale = scale,
        c_contrasts = c_contrasts, contrasts = contrasts,
        n_eigen = n_eigen, n_medoids = n_medoids,
        eigdecomp_tol = eigdecomp_tol, eigdecomp_iter = eigdecomp_iter
      )
    } else {
      opt_params <- bpFitGrid(
        target = target, center = center, scale = scale,
        alg = alg, c_contrasts = c_contrasts,
        contrasts = contrasts, penalties = penalties,
        n_eigen = n_eigen, clust_method = clust_method,
        n_centers = n_centers, max_iter = max_iter,
        linkage_method = linkage_method, clusters = clusters,
        eigdecomp_tol = eigdecomp_tol, eigdecomp_iter = eigdecomp_iter
      )
    }
  }
  return(opt_params)
}

###############################################################################

#' Fold-Specific Selection of Contrastive and Penalization Parameters
#'
#' @description A wrapper function for fitting various internal functions to
#'  select the optimal setting of the contrastive and penalization parameters
#'  via cross-validation. For internal use only.
#'
#' @param fold Object specifying cross-validation folds as generated by a call
#'  to \code{\link[origami]{make_folds}}.
#' @param target The target (experimental) data set, in a standard format such
#'  as a \code{data.frame} or \code{matrix}.
#' @param background The background data set, in a standard format such as a
#'  \code{data.frame} or \code{matrix}. Note that the number of features must
#'  match the number of features in the target data.
#' @param center A \code{logical} indicating whether the target and background
#'  data sets should be centered to mean zero.
#' @param scale A \code{logical} indicating whether the target and background
#'  data sets should be scaled to unit variance.
#' @param n_eigen A \code{numeric} indicating the number of eigenvectors (or
#'  sparse contrastive components) to be computed. The default is to compute
#'  two such eigenvectors.
#' @param alg A \code{character} indicating the SPCA algorithm used to sparsify
#'  the contrastive loadings. Currently supports \code{iterative} for the
#'  \insertCite{zou2006sparse;textual}{scPCA} implementation, \code{var_proj}
#'  for the non-randomized \insertCite{erichson2018sparse;textual}{scPCA}
#'  solution, and \code{rand_var_proj} for the randomized
#'  \insertCite{erichson2018sparse;textual}{scPCA} result.
#' @param contrasts A \code{numeric} vector of the contrastive parameters. Each
#'  element must be a unique non-negative real number. The default is to use 40
#'  logarithmically spaced values between 0.1 and 1000.
#' @param penalties A \code{numeric} vector of the L1 penalty terms on the
#'  loadings. The default is to use 20 equidistant values between 0.05 and 1.
#' @param clust_method A \code{character} specifying the clustering method to
#'  use for choosing the optimal contrastive parameter. Currently, this is
#'  limited to either k-means, partitioning around medoids (PAM), and
#'  hierarchical clustering. The default is k-means clustering.
#' @param n_centers A \code{numeric} giving the number of centers to use in the
#'  clustering algorithm. If set to 1, cPCA, as first proposed by Abid et al.,
#'  is performed, regardless of what the \code{penalties} argument is set to.
#' @param max_iter A \code{numeric} giving the maximum number of iterations to
#'   be used in k-means clustering, defaulting to 10.
#' @param linkage_method A \code{character} specifying the agglomerative
#'   linkage method to be used if \code{clust_method = "hclust"}. The options
#'   are \code{ward.D2}, \code{single}, \code{complete}, \code{average},
#'   \code{mcquitty}, \code{median}, and \code{centroid}. The default is
#'   \code{complete}.
#' @param n_medoids A \code{numeric} indicating the number of medoids to
#'  consider if \code{n_centers} is set to 1. The default is 8 such medoids.
#' @param parallel A \code{logical} indicating whether to invoke parallel
#'  processing via the \pkg{BiocParallel} infrastructure. The default is
#'  \code{FALSE} for sequential evaluation.
#' @param clusters A \code{numeric} vector of cluster labels for observations in
#'  the \code{target} data. Defaults to \code{NULL}, but is otherwise used to
#'  identify the optimal set of hyperparameters when fitting the scPCA and the
#'  automated version of cPCA.
#' @param eigdecomp_tol A \code{numeric} providing the level of precision used by
#'  eigendecompositon calculations. Defaults to \code{1e-10}.
#' @param eigdecomp_iter A \code{numeric} indicating the maximum number of
#'  interations performed by eigendecompositon calculations. Defaults to
#'  \code{1000}.
#'
#' @importFrom origami training validation
#'
#' @return Output structure matching either that of \code{\link{fitCPCA}} or
#'  \code{\link{fitGrid}} (or their parallelized variants, namely either
#'  \code{\link{bpFitCPCA}} and \code{link{bpFitGrid}}, respectively).
#'
#' @references
#'   \insertAllCited{}
#'   
#' @keywords internal
cvSelectParams <- function(fold, target, background, center, scale, n_eigen,
                           alg = alg, contrasts, penalties, clust_method,
                           n_centers, max_iter, linkage_method, n_medoids,
                           parallel, clusters, eigdecomp_tol, eigdecomp_iter) {
  
  # make training and validation folds
  train_target <- origami::training(target, fold$target)
  valid_target <- origami::validation(target, fold$target)
  train_background <- origami::training(background, fold$background)

  # call parallelized function variants if so requested
  if (!parallel) {
    # create contrastive covariance matrices
    c_contrasts <- contrastiveCov(
      target = train_target, background = train_background,
      contrasts = contrasts, center = center, scale = scale
    )
    if (n_centers == 1) {
      opt_params <- fitCPCA(
        target = train_target, center = center,
        scale = scale, c_contrasts = c_contrasts,
        contrasts = contrasts, n_eigen = n_eigen,
        n_medoids = n_medoids, eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter =  eigdecomp_iter
      )
    } else {
      opt_params <- fitGrid(
        target = train_target,
        target_valid = valid_target,
        center = center, scale = scale,
        alg = alg,
        c_contrasts = c_contrasts,
        contrasts = contrasts,
        penalties = penalties,
        n_eigen = n_eigen,
        clust_method = clust_method,
        n_centers = n_centers,
        max_iter = max_iter,
        linkage_method = linkage_method,
        clusters = clusters,
        eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter =  eigdecomp_iter
      )
    }
  } else {
    # create contrastive covariance matrices
    c_contrasts <- bpContrastiveCov(
      target = train_target, background = train_background,
      contrasts = contrasts, center = center, scale = scale
    )
    if (n_centers == 1) {
      opt_params <- bpFitCPCA(
        target = train_target, center = center,
        scale = scale, c_contrasts = c_contrasts,
        contrasts = contrasts, n_eigen = n_eigen,
        n_medoids = n_medoids, eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter =  eigdecomp_iter
      )
    } else {
      opt_params <- bpFitGrid(
        target = train_target, center = center,
        scale = scale, c_contrasts = c_contrasts,
        contrasts = contrasts, penalties = penalties,
        n_eigen = n_eigen,
        alg = alg,
        clust_method = clust_method,
        target_valid = valid_target,
        n_centers = n_centers,
        max_iter = max_iter,
        linkage_method = linkage_method,
        clusters = clusters,
        eigdecomp_tol = eigdecomp_tol,
        eigdecomp_iter =  eigdecomp_iter
      )
    }
  }
  return(opt_params)
}