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wip: proper package implementation

This commit is contained in:
Daniel Kapla 2019-08-12 00:33:52 +02:00
parent 095e463463
commit b071a689d9
17 changed files with 491 additions and 132 deletions

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@ -1,10 +1,18 @@
# Generated by roxygen2: do not edit by hand # Generated by roxygen2: do not edit by hand
S3method(plot,cve)
export(cve) export(cve)
export(cve_cpp) export(cve.call)
export(dataset) export(dataset)
export(estimateBandwidth) export(estimateBandwidth)
export(rStiefel) export(rStiefel)
import(Rcpp) import(Rcpp)
import(stats)
importFrom(Rcpp,evalCpp) importFrom(Rcpp,evalCpp)
importFrom(graphics,lines)
importFrom(graphics,plot)
importFrom(graphics,points)
importFrom(stats,model.frame)
importFrom(stats,rbinom)
importFrom(stats,rnorm)
useDynLib(CVE) useDynLib(CVE)

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@ -1,44 +1,145 @@
#' Conditional Variance Estimator #' Implementation of the CVE method.
#' #'
#' Conditional Variance Estimator (CVE) is a novel sufficient dimension #' Conditional Variance Estimator (CVE) is a novel sufficient dimension
#' reduction (SDR) method for regressions satisfying E(Y|X) = E(Y|B'X), #' reduction (SDR) method assuming a model
#' where B'X is a lower dimensional projection of the predictors. #' \deqn{Y \sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
#' where B'X is a lower dimensional projection of the predictors.
#' #'
#' @param X A matrix of type numeric of dimensions N times dim where N is the number #' @param formula Formel for the regression model defining `X`, `Y`.
#' of entries with dim as data dimension. #' See: \code{\link{formula}}.
#' @param Y A vector of type numeric of length N (coresponds to \code{x}). #' @param data data.frame holding data for formula.
#' @param k Guess for rank(B). #' @param method The different only differe in the used optimization.
#' @param nObs As describet in the paper. #' All of them are Gradient based optimization on a Stiefel manifold.
#' \itemize{
#' \item "simple" Simple reduction of stepsize.
#' \item "linesearch" determines stepsize with backtracking linesearch
#' using Armijo-Wolf conditions.
#' \item TODO: further
#' }
#' @param ... Further parameters depending on the used method.
#' TODO: See ...
#' @examples
#' library(CVE)
#' ds <- dataset("M5")
#' X <- ds$X
#' Y <- ds$Y
#' dr <- cve(Y ~ X, k = 1)
#' #'
#' @param tol Tolerance for optimization stopping creteria. #' @references Fertl L, Bura E. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
#'
#' @import stats
#' @importFrom stats model.frame
#' @export #' @export
cve <- function(formula, data, method = "simple", ...) {
# check for type of `data` if supplied and set default
if (missing(data)) {
data <- environment(formula)
} else if (!is.data.frame(data)) {
stop('Parameter `data` must be a `data.frame` or missing.')
}
# extract `X`, `Y` from `formula` with `data`
model <- stats::model.frame(formula, data)
X <- as.matrix(model[,-1, drop=FALSE])
Y <- as.matrix(model[, 1, drop=FALSE])
# pass extracted data on to [cve.call()]
dr <- cve.call(X, Y, method = method, ...)
# overwrite `call` property from [cve.call()]
dr$call <- match.call()
return(dr)
}
#' @rdname cve
#' @export
cve.call <- function(X, Y, method = "simple", nObs = nrow(X)^.5, k, ...) {
# TODO: replace default value of `k` by `max.dim` when fast enough
if (missing(k)) {
stop("TODO: parameter `k` (rank(B)) is required, replace by `max.dim`.")
}
# parameter checking
if (!(is.matrix(X) && is.matrix(Y))) {
stop('X and Y should be matrices.')
}
if (nrow(X) != nrow(Y)) {
stop('Rows of X and Y are not compatible.')
}
if (ncol(X) < 2) {
stop('X is one dimensional, no need for dimension reduction.')
}
if (ncol(Y) > 1) {
stop('Only one dimensional responces Y are supported.')
}
# call C++ CVE implementation
# dr ... Dimension Reduction
dr <- cve_cpp(X, Y, tolower(method), k = k, nObs = nObs, ...)
# augment result information
dr$method <- method
dr$call <- match.call()
class(dr) <- "cve"
return(dr)
}
# TODO: write summary
# summary.cve <- function() {
# # code #
# }
#' Ploting helper for objects of class \code{cve}.
#' #'
#' @seealso TODO: \code{vignette("CVE_paper", package="CVE")}. #' @param x Object of class \code{cve} (result of [cve()]).
#' @param content Specifies what to plot:
#' \itemize{
#' \item "history" Plots the loss history from stiefel optimization
#' (default).
#' \item ... TODO: add (if there are any)
#' }
#' @param ... Pass through parameters to [plot()] and [lines()]
#' #'
#' @references Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019 #' @seealso see \code{\link{par}} for graphical parameters to pass through.
cve <- function(X, Y, k, #' @importFrom graphics plot lines points
nObs = sqrt(nrow(X)), #' @method plot cve
tauInitial = 1.0, #' @export
tol = 1e-3, plot.cve <- function(x, ...) {
slack = -1e-10,
maxIter = 50L, H <- x$history
attempts = 10L H_1 <- H[H[, 1] > 0, 1]
) {
# check data parameter types defaults <- list(
stopifnot( main = "History",
is.matrix(X), xlab = "Iterations i",
is.vector(Y), ylab = expression(loss == L[n](V^{(i)})),
typeof(X) == 'double', xlim = c(1, nrow(H)),
typeof(Y) == 'double' ylim = c(0, max(H)),
type = "l"
) )
# call CVE C++ implementation call.plot <- match.call()
return(cve_cpp(X, Y, k, keys <- names(defaults)
nObs, keys <- keys[match(keys, names(call.plot)[-1], nomatch = 0) == 0]
tauInitial,
tol, for (key in keys) {
slack, call.plot[[key]] <- defaults[[key]]
maxIter, }
attempts
)) call.plot[[1L]] <- quote(plot)
call.plot$x <- quote(1:length(H_1))
call.plot$y <- quote(H_1)
eval(call.plot)
if (ncol(H) > 1) {
for (i in 2:ncol(H)) {
H_i <- H[H[, i] > 0, i]
lines(1:length(H_i), H_i)
}
}
x.ends <- apply(H, 2, function(h) { length(h[h > 0]) })
y.ends <- apply(H, 2, function(h) { min(h[h > 0]) })
points(x.ends, y.ends)
} }

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@ -1,6 +1,18 @@
# Generated by using Rcpp::compileAttributes() -> do not edit by hand # Generated by using Rcpp::compileAttributes() -> do not edit by hand
# Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393 # Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393
#' Gradient computation of the loss `L_n(V)`.
#'
#' The loss is defined as
#' \deqn{L_n(V) := \frac{1}{n}\sum_{j=1}^n y_2(V, X_j) - y_1(V, X_j)^2}{L_n(V) := 1/n sum_j( (y_2(V, X_j) - y_1(V, X_j)^2 )}
#' with
#' \deqn{y_l(s_0) := \sum_{i=1}^n w_i(V, s_0)Y_i^l}{y_l(s_0) := sum_i(w_i(V, s_0) Y_i^l)}
#'
#' @rdname optStiefel
#' @keywords internal
#' @name gradient
NULL
#' Stiefel Optimization. #' Stiefel Optimization.
#' #'
#' Stiefel Optimization for \code{V} given a dataset \code{X} and responces #' Stiefel Optimization for \code{V} given a dataset \code{X} and responces
@ -24,8 +36,13 @@
#' orthogonal space spaned by \code{V}. #' orthogonal space spaned by \code{V}.
#' #'
#' @rdname optStiefel #' @rdname optStiefel
#' @name optStiefel
#' @keywords internal #' @keywords internal
#' @name optStiefel_simple
NULL
#' @rdname optStiefel
#' @keywords internal
#' @name optStiefel_linesearch
NULL NULL
#' Estimated bandwidth for CVE. #' Estimated bandwidth for CVE.
@ -87,9 +104,8 @@ rStiefel <- function(p, q) {
#' and the matrix \code{B} estimated for the model as a orthogonal basis of the #' and the matrix \code{B} estimated for the model as a orthogonal basis of the
#' orthogonal space spaned by \code{V}. #' orthogonal space spaned by \code{V}.
#' #'
#' @rdname cve_cpp_V1 #' @keywords internal
#' @export cve_cpp <- function(X, Y, method, k, nObs, tauInitial = 1., rho1 = 0.1, rho2 = 0.9, tol = 1e-5, maxIter = 50L, maxLineSearchIter = 10L, attempts = 10L) {
cve_cpp <- function(X, Y, k, nObs, tauInitial = 1., tol = 1e-5, slack = -1e-10, maxIter = 50L, attempts = 10L) { .Call('_CVE_cve_cpp', PACKAGE = 'CVE', X, Y, method, k, nObs, tauInitial, rho1, rho2, tol, maxIter, maxLineSearchIter, attempts)
.Call('_CVE_cve_cpp', PACKAGE = 'CVE', X, Y, k, nObs, tauInitial, tol, slack, maxIter, attempts)
} }

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@ -12,10 +12,12 @@
#' @param lambda Only for \code{"M4"}, see: below. #' @param lambda Only for \code{"M4"}, see: below.
#' #'
#' @return List with elements #' @return List with elements
#' \item{X}{data} #' \itemize{
#' \item{Y}{response} #' \item{X}{data}
#' \item{B}{Used dim-reduction matrix} #' \item{Y}{response}
#' \item{name}{Name of the dataset (name parameter)} #' \item{B}{Used dim-reduction matrix}
#' \item{name}{Name of the dataset (name parameter)}
#' }
#' #'
#' @section M1: #' @section M1:
#' The data follows \eqn{X\sim N_p(0, \Sigma)}{X ~ N_p(0, Sigma)} for a subspace #' The data follows \eqn{X\sim N_p(0, \Sigma)}{X ~ N_p(0, Sigma)} for a subspace
@ -33,9 +35,9 @@
#' @section M5: #' @section M5:
#' TODO: #' TODO:
#' #'
#' @import stats
#' @importFrom stats rnorm rbinom
#' @export #' @export
#'
#' @references Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
dataset <- function(name = "M1", n, B, p.mix = 0.3, lambda = 1.0) { dataset <- function(name = "M1", n, B, p.mix = 0.3, lambda = 1.0) {
# validate parameters # validate parameters
stopifnot(name %in% c("M1", "M2", "M3", "M4", "M5")) stopifnot(name %in% c("M1", "M2", "M3", "M4", "M5"))

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@ -5,6 +5,9 @@
#' #'
#' TODO: And some details #' TODO: And some details
#' #'
#'
#' @references Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
#'
#' @docType package #' @docType package
#' @author Loki #' @author Loki
#' @import Rcpp #' @import Rcpp

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@ -12,6 +12,9 @@ Conditional Variance Estimator for Sufficient Dimension
\details{ \details{
TODO: And some details TODO: And some details
} }
\references{
Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
}
\author{ \author{
Loki Loki
} }

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@ -2,31 +2,45 @@
% Please edit documentation in R/CVE.R % Please edit documentation in R/CVE.R
\name{cve} \name{cve}
\alias{cve} \alias{cve}
\title{Conditional Variance Estimator} \alias{cve.call}
\title{Implementation of the CVE method.}
\usage{ \usage{
cve(X, Y, k, nObs = sqrt(nrow(X)), tauInitial = 1, tol = 0.001, cve(formula, data, method = "simple", ...)
slack = -1e-10, maxIter = 50L, attempts = 10L)
cve.call(X, Y, method = "simple", nObs = nrow(X)^0.5, k, ...)
} }
\arguments{ \arguments{
\item{X}{A matrix of type numeric of dimensions N times dim where N is the number \item{formula}{Formel for the regression model defining `X`, `Y`.
of entries with dim as data dimension.} See: \code{\link{formula}}.}
\item{Y}{A vector of type numeric of length N (coresponds to \code{x}).} \item{data}{data.frame holding data for formula.}
\item{k}{Guess for rank(B).} \item{method}{The different only differe in the used optimization.
All of them are Gradient based optimization on a Stiefel manifold.
\itemize{
\item "simple" Simple reduction of stepsize.
\item "linesearch" determines stepsize with backtracking linesearch
using Armijo-Wolf conditions.
\item TODO: further
}}
\item{nObs}{As describet in the paper.} \item{...}{Further parameters depending on the used method.
TODO: See ...}
\item{tol}{Tolerance for optimization stopping creteria.}
} }
\description{ \description{
Conditional Variance Estimator (CVE) is a novel sufficient dimension Conditional Variance Estimator (CVE) is a novel sufficient dimension
reduction (SDR) method for regressions satisfying E(Y|X) = E(Y|B'X), reduction (SDR) method assuming a model
where B'X is a lower dimensional projection of the predictors. \deqn{Y \sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
where B'X is a lower dimensional projection of the predictors.
}
\examples{
library(CVE)
ds <- dataset("M5")
X <- ds$X
Y <- ds$Y
dr <- cve(Y ~ X, k = 1)
} }
\references{ \references{
Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019 Fertl L, Bura E. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
}
\seealso{
TODO: \code{vignette("CVE_paper", package="CVE")}.
} }

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@ -4,8 +4,9 @@
\alias{cve_cpp} \alias{cve_cpp}
\title{Conditional Variance Estimation (CVE) method.} \title{Conditional Variance Estimation (CVE) method.}
\usage{ \usage{
cve_cpp(X, Y, k, nObs, tauInitial = 1, tol = 1e-05, slack = -1e-10, cve_cpp(X, Y, method, k, nObs, tauInitial = 1, rho1 = 0.1,
maxIter = 50L, attempts = 10L) rho2 = 0.9, tol = 1e-05, maxIter = 50L, maxLineSearchIter = 10L,
attempts = 10L)
} }
\arguments{ \arguments{
\item{X}{data points} \item{X}{data points}
@ -19,13 +20,13 @@ cve_cpp(X, Y, k, nObs, tauInitial = 1, tol = 1e-05, slack = -1e-10,
\item{tol}{Tolerance for update error used for stopping criterion (default 1e-5)} \item{tol}{Tolerance for update error used for stopping criterion (default 1e-5)}
\item{slack}{Ratio of small negative error allowed in loss optimization (default -1e-10)}
\item{maxIter}{Upper bound of optimization iterations (default 50)} \item{maxIter}{Upper bound of optimization iterations (default 50)}
\item{attempts}{Number of tryes with new random optimization starting points (default 10)} \item{attempts}{Number of tryes with new random optimization starting points (default 10)}
\item{tau}{Initial step size (default 1)} \item{tau}{Initial step size (default 1)}
\item{slack}{Ratio of small negative error allowed in loss optimization (default -1e-10)}
} }
\value{ \value{
List containing the bandwidth \code{h}, optimization objective \code{V} List containing the bandwidth \code{h}, optimization objective \code{V}
@ -35,3 +36,4 @@ List containing the bandwidth \code{h}, optimization objective \code{V}
\description{ \description{
This version uses a "simple" stiefel optimization schema. This version uses a "simple" stiefel optimization schema.
} }
\keyword{internal}

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@ -19,10 +19,12 @@ dataset(name = "M1", n, B, p.mix = 0.3, lambda = 1)
} }
\value{ \value{
List with elements List with elements
\item{X}{data} \itemize{
\item{Y}{response} \item{X}{data}
\item{B}{Used dim-reduction matrix} \item{Y}{response}
\item{name}{Name of the dataset (name parameter)} \item{B}{Used dim-reduction matrix}
\item{name}{Name of the dataset (name parameter)}
}
} }
\description{ \description{
Provides sample datasets. There are 5 different datasets named Provides sample datasets. There are 5 different datasets named
@ -60,6 +62,3 @@ TODO:
TODO: TODO:
} }
\references{
Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
}

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@ -1,8 +1,10 @@
% Generated by roxygen2: do not edit by hand % Generated by roxygen2: do not edit by hand
% Please edit documentation in R/RcppExports.R % Please edit documentation in R/RcppExports.R
\name{optStiefel} \name{gradient}
\alias{optStiefel} \alias{gradient}
\title{Stiefel Optimization.} \alias{optStiefel_simple}
\alias{optStiefel_linesearch}
\title{Gradient computation of the loss `L_n(V)`.}
\arguments{ \arguments{
\item{X}{data points} \item{X}{data points}
@ -27,6 +29,11 @@ List containing the bandwidth \code{h}, optimization objective \code{V}
orthogonal space spaned by \code{V}. orthogonal space spaned by \code{V}.
} }
\description{ \description{
The loss is defined as
\deqn{L_n(V) := \frac{1}{n}\sum_{j=1}^n y_2(V, X_j) - y_1(V, X_j)^2}{L_n(V) := 1/n sum_j( (y_2(V, X_j) - y_1(V, X_j)^2 )}
with
\deqn{y_l(s_0) := \sum_{i=1}^n w_i(V, s_0)Y_i^l}{y_l(s_0) := sum_i(w_i(V, s_0) Y_i^l)}
Stiefel Optimization for \code{V} given a dataset \code{X} and responces Stiefel Optimization for \code{V} given a dataset \code{X} and responces
\code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon} \code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{% to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%

25
CVE/man/plot.cve.Rd Normal file
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@ -0,0 +1,25 @@
% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/CVE.R
\name{plot.cve}
\alias{plot.cve}
\title{Ploting helper for objects of class \code{"cve"}.}
\usage{
\method{plot}{cve}(x, ...)
}
\arguments{
\item{x}{Object of class \code{"cve"} (result of [cve()]).}
\item{...}{Pass through parameters to [plot()] and [lines()]}
\item{content}{Specifies what to plot:
\itemize{
\item "history" Plots the loss history from stiefel optimization.
\item ... TODO: add (if there are any)
}}
}
\description{
Ploting helper for objects of class \code{"cve"}.
}
\seealso{
see \code{\link{par}} for graphical parameters to pass through.
}

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@ -1,8 +1,3 @@
//
// Usage:
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')"
//
// only `RcppArmadillo.h` which includes `Rcpp.h` // only `RcppArmadillo.h` which includes `Rcpp.h`
#include <RcppArmadillo.h> #include <RcppArmadillo.h>
@ -14,6 +9,9 @@
// required for `R::qchisq()` used in `estimateBandwidth()` // required for `R::qchisq()` used in `estimateBandwidth()`
#include <Rmath.h> #include <Rmath.h>
// for proper error handling
#include <stdexcept>
//' Estimated bandwidth for CVE. //' Estimated bandwidth for CVE.
//' //'
//' Estimates a propper bandwidth \code{h} according //' Estimates a propper bandwidth \code{h} according
@ -72,18 +70,27 @@ arma::mat rStiefel(arma::uword p, arma::uword q) {
return Q; return Q;
} }
//' Gradient computation of the loss `L_n(V)`.
//'
//' The loss is defined as
//' \deqn{L_n(V) := \frac{1}{n}\sum_{j=1}^n y_2(V, X_j) - y_1(V, X_j)^2}{L_n(V) := 1/n sum_j( (y_2(V, X_j) - y_1(V, X_j)^2 )}
//' with
//' \deqn{y_l(s_0) := \sum_{i=1}^n w_i(V, s_0)Y_i^l}{y_l(s_0) := sum_i(w_i(V, s_0) Y_i^l)}
//'
//' @rdname optStiefel
//' @keywords internal
//' @name gradient
double gradient(const arma::mat& X, double gradient(const arma::mat& X,
const arma::mat& X_diff, const arma::mat& X_diff,
const arma::mat& Y, const arma::mat& Y,
const arma::mat& Y_rep, const arma::mat& Y_rep,
const arma::mat& V, const arma::mat& V,
const double h, const double h,
arma::mat* G = 0 arma::mat* G = 0 // out
) { ) {
using namespace arma; using namespace arma;
uword n = X.n_rows; uword n = X.n_rows;
uword p = X.n_cols;
// orthogonal projection matrix `Q = I - VV'` for dist computation // orthogonal projection matrix `Q = I - VV'` for dist computation
mat Q = -(V * V.t()); mat Q = -(V * V.t());
@ -138,18 +145,17 @@ double gradient(const arma::mat& X,
//' orthogonal space spaned by \code{V}. //' orthogonal space spaned by \code{V}.
//' //'
//' @rdname optStiefel //' @rdname optStiefel
//' @name optStiefel
//' @keywords internal //' @keywords internal
double optStiefel( //' @name optStiefel_simple
double optStiefel_simple(
const arma::mat& X, const arma::mat& X,
const arma::vec& Y, const arma::vec& Y,
const int k, const int k,
const double h, const double h,
const double tauInitial, const double tauInitial,
const double tol, const double tol,
const double slack,
const int maxIter, const int maxIter,
arma::mat& V, // out arma::mat& V, // out
arma::vec& history // out arma::vec& history // out
) { ) {
using namespace arma; using namespace arma;
@ -177,9 +183,9 @@ double optStiefel(
double loss; double loss;
double error = datum::inf; double error = datum::inf;
double tau = tauInitial; double tau = tauInitial;
int count; int iter;
// main optimization loop // main optimization loop
for (count = 0; count < maxIter && error > tol; ++count) { for (iter = 0; iter < maxIter && error > tol; ++iter) {
// calc gradient `G = grad_V(L)(V)` // calc gradient `G = grad_V(L)(V)`
mat G; mat G;
loss = gradient(X, X_diff, Y, Y_rep, V, h, &G); loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
@ -190,11 +196,11 @@ double optStiefel(
// loss after step `L(V(tau))` // loss after step `L(V(tau))`
double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h); double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h);
// store `loss` in `history` and increase `count` // store `loss` in `history` and increase `iter`
history(count) = loss; history(iter) = loss;
// validate if loss decreased // validate if loss decreased
if ((loss_tau - loss) > slack * loss) { if ((loss_tau - loss) > 0.0) {
// ignore step, retry with half the step size // ignore step, retry with half the step size
tau = tau / 2.; tau = tau / 2.;
error = datum::inf; error = datum::inf;
@ -208,7 +214,121 @@ double optStiefel(
} }
// store final `loss` // store final `loss`
history(count) = loss; history(iter) = loss;
return loss;
}
//' @rdname optStiefel
//' @keywords internal
//' @name optStiefel_linesearch
double optStiefel_linesearch(
const arma::mat& X,
const arma::vec& Y,
const int k,
const double h,
const double tauInitial,
const double tol,
const int maxIter,
const double rho1,
const double rho2,
const int maxLineSearchIter,
arma::mat& V, // out
arma::vec& history // out
) {
using namespace arma;
// get dimensions
const uword n = X.n_rows; // nr samples
const uword p = X.n_cols; // dim of random variable `X`
const uword q = p - k; // rank(V) e.g. dim of ortho space of span{B}
// all `X_i - X_j` differences, `X_diff.row(i * n + j) = X_i - X_j`
mat X_diff(n * n, p);
for (uword i = 0, k = 0; i < n; ++i) {
for (uword j = 0; j < n; ++j) {
X_diff.row(k++) = X.row(i) - X.row(j);
}
}
const vec Y_rep = repmat(Y, n, 1);
const mat I_p = eye<mat>(p, p);
const mat I_2q = eye<mat>(2 * q, 2 * q);
// initial start value for `V`
V = rStiefel(p, q);
// first gradient initialization
mat G;
double loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
// set first `loss` in history
history(0) = loss;
// main curvilinear optimization loop
double error = datum::inf;
int iter = 0;
while (iter++ < maxIter && error > tol) {
// helper matrices `lU` (linesearch U), `lV` (linesearch V)
// as describet in [Wen, Yin] Lemma 4.
mat lU = join_rows(G, V); // linesearch "U"
mat lV = join_rows(V, -1.0 * G); // linesearch "V"
// `A = G V' - V G'`
mat A = lU * lV.t();
// set initial step size for curvilinear line search
double tau = tauInitial, lower = 0., upper = datum::inf;
// TODO: check if `tau` is valid for inverting
// set line search internal gradient and loss to cycle for next iteration
mat V_tau; // next position after a step of size `tau`, a.k.a. `V(tau)`
mat G_tau; // gradient of `V` at `V(tau) = V_tau`
double loss_tau; // loss (objective) at position `V(tau)`
int lsIter = 0; // linesearch iter
// start line search
do {
mat HV = inv(I_2q + (tau/2.) * lV.t() * lU) * lV.t();
// next step `V`
V_tau = V - tau * (lU * (HV * V));
double LprimeV = -trace(G.t() * A * V);
mat lB = I_p - (tau / 2.) * lU * HV;
loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h, &G_tau);
double LprimeV_tau = -2. * trace(G_tau.t() * lB * A * (V + V_tau));
// Armijo condition
if (loss_tau > loss + (rho1 * tau * LprimeV)) {
upper = tau;
tau = (lower + upper) / 2.;
// Wolfe condition
} else if (LprimeV_tau < rho2 * LprimeV) {
lower = tau;
if (upper == datum::inf) {
tau = 2. * lower;
} else {
tau = (lower + upper) / 2.;
}
} else {
break;
}
} while (++lsIter < maxLineSearchIter);
// compute error (break condition)
// Note: `error` is the decrease of the objective `L_n(V)` and not the
// norm of the gradient as proposed in [Wen, Yin] Algorithm 1.
error = loss - loss_tau;
// cycle `V`, `G` and `loss` for next iteration
V = V_tau;
loss = loss_tau;
G = G_tau;
// store final `loss`
history(iter) = loss;
}
return loss; return loss;
} }
@ -232,18 +352,20 @@ double optStiefel(
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the //' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}. //' orthogonal space spaned by \code{V}.
//' //'
//' @rdname cve_cpp_V1 //' @keywords internal
//' @export
// [[Rcpp::export]] // [[Rcpp::export]]
Rcpp::List cve_cpp( Rcpp::List cve_cpp(
const arma::mat& X, const arma::mat& X,
const arma::vec& Y, const arma::vec& Y,
const std::string method,
const int k, const int k,
const double nObs, const double nObs,
const double tauInitial = 1., const double tauInitial = 1.,
const double rho1 = 0.1,
const double rho2 = 0.9,
const double tol = 1e-5, const double tol = 1e-5,
const double slack = -1e-10,
const int maxIter = 50, const int maxIter = 50,
const int maxLineSearchIter = 10,
const int attempts = 10 const int attempts = 10
) { ) {
using namespace arma; using namespace arma;
@ -263,12 +385,27 @@ Rcpp::List cve_cpp(
// declare output variables // declare output variables
mat V; // estimated `V` space mat V; // estimated `V` space
vec hist = vec(history.n_rows, fill::zeros); // optimization history vec hist = vec(history.n_rows, fill::zeros); // optimization history
double loss = optStiefel(X, Y, k, h, tauInitial, tol, slack, maxIter, V, hist); double loss;
if (method == "simple") {
loss = optStiefel_simple(
X, Y, k, h,
tauInitial, tol, maxIter,
V, hist
);
} else if (method == "linesearch") {
loss = optStiefel_linesearch(
X, Y, k, h,
tauInitial, tol, maxIter, rho1, rho2, maxLineSearchIter,
V, hist
);
} else {
throw std::invalid_argument("Unknown method.");
}
if (loss < loss_best) { if (loss < loss_best) {
loss_best = loss; loss_best = loss;
V_best = V; V_best = V;
} }
// write history to history collection // add history to history collection
history.col(i) = hist; history.col(i) = hist;
} }

View File

@ -32,21 +32,24 @@ BEGIN_RCPP
END_RCPP END_RCPP
} }
// cve_cpp // cve_cpp
Rcpp::List cve_cpp(const arma::mat& X, const arma::vec& Y, const int k, const double nObs, const double tauInitial, const double tol, const double slack, const int maxIter, const int attempts); Rcpp::List cve_cpp(const arma::mat& X, const arma::vec& Y, const std::string method, const int k, const double nObs, const double tauInitial, const double rho1, const double rho2, const double tol, const int maxIter, const int maxLineSearchIter, const int attempts);
RcppExport SEXP _CVE_cve_cpp(SEXP XSEXP, SEXP YSEXP, SEXP kSEXP, SEXP nObsSEXP, SEXP tauInitialSEXP, SEXP tolSEXP, SEXP slackSEXP, SEXP maxIterSEXP, SEXP attemptsSEXP) { RcppExport SEXP _CVE_cve_cpp(SEXP XSEXP, SEXP YSEXP, SEXP methodSEXP, SEXP kSEXP, SEXP nObsSEXP, SEXP tauInitialSEXP, SEXP rho1SEXP, SEXP rho2SEXP, SEXP tolSEXP, SEXP maxIterSEXP, SEXP maxLineSearchIterSEXP, SEXP attemptsSEXP) {
BEGIN_RCPP BEGIN_RCPP
Rcpp::RObject rcpp_result_gen; Rcpp::RObject rcpp_result_gen;
Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::RNGScope rcpp_rngScope_gen;
Rcpp::traits::input_parameter< const arma::mat& >::type X(XSEXP); Rcpp::traits::input_parameter< const arma::mat& >::type X(XSEXP);
Rcpp::traits::input_parameter< const arma::vec& >::type Y(YSEXP); Rcpp::traits::input_parameter< const arma::vec& >::type Y(YSEXP);
Rcpp::traits::input_parameter< const std::string >::type method(methodSEXP);
Rcpp::traits::input_parameter< const int >::type k(kSEXP); Rcpp::traits::input_parameter< const int >::type k(kSEXP);
Rcpp::traits::input_parameter< const double >::type nObs(nObsSEXP); Rcpp::traits::input_parameter< const double >::type nObs(nObsSEXP);
Rcpp::traits::input_parameter< const double >::type tauInitial(tauInitialSEXP); Rcpp::traits::input_parameter< const double >::type tauInitial(tauInitialSEXP);
Rcpp::traits::input_parameter< const double >::type rho1(rho1SEXP);
Rcpp::traits::input_parameter< const double >::type rho2(rho2SEXP);
Rcpp::traits::input_parameter< const double >::type tol(tolSEXP); Rcpp::traits::input_parameter< const double >::type tol(tolSEXP);
Rcpp::traits::input_parameter< const double >::type slack(slackSEXP);
Rcpp::traits::input_parameter< const int >::type maxIter(maxIterSEXP); Rcpp::traits::input_parameter< const int >::type maxIter(maxIterSEXP);
Rcpp::traits::input_parameter< const int >::type maxLineSearchIter(maxLineSearchIterSEXP);
Rcpp::traits::input_parameter< const int >::type attempts(attemptsSEXP); Rcpp::traits::input_parameter< const int >::type attempts(attemptsSEXP);
rcpp_result_gen = Rcpp::wrap(cve_cpp(X, Y, k, nObs, tauInitial, tol, slack, maxIter, attempts)); rcpp_result_gen = Rcpp::wrap(cve_cpp(X, Y, method, k, nObs, tauInitial, rho1, rho2, tol, maxIter, maxLineSearchIter, attempts));
return rcpp_result_gen; return rcpp_result_gen;
END_RCPP END_RCPP
} }
@ -54,7 +57,7 @@ END_RCPP
static const R_CallMethodDef CallEntries[] = { static const R_CallMethodDef CallEntries[] = {
{"_CVE_estimateBandwidth", (DL_FUNC) &_CVE_estimateBandwidth, 3}, {"_CVE_estimateBandwidth", (DL_FUNC) &_CVE_estimateBandwidth, 3},
{"_CVE_rStiefel", (DL_FUNC) &_CVE_rStiefel, 2}, {"_CVE_rStiefel", (DL_FUNC) &_CVE_rStiefel, 2},
{"_CVE_cve_cpp", (DL_FUNC) &_CVE_cve_cpp, 9}, {"_CVE_cve_cpp", (DL_FUNC) &_CVE_cve_cpp, 12},
{NULL, NULL, 0} {NULL, NULL, 0}
}; };

View File

@ -0,0 +1,7 @@
# Overview
- **CVE/**: Contains actual `R` package.
- **CVE_legacy/**: Contains original (first) `R` implementatin of the CVE method.
The `*.R` and `*.cpp` files in the root directory are _development_ and _test_ files.
## TODO: README.md

View File

@ -1,5 +1,5 @@
// //
// Standalone implementation for development. // Development file.
// //
// Usage: // Usage:
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')" // ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')"
@ -74,6 +74,15 @@ arma::mat rStiefel(arma::uword p, arma::uword q) {
return Q; return Q;
} }
//' Gradient computation of the loss `L_n(V)`.
//'
//' The loss is defined as
//' \deqn{L_n(V) := \frac{1}{n}\sum_{j=1}^n (y_2(V, X_j) - y_1(V, X_j)^2)}
//' with
//' \deqn{y_l(s_0) := \sum_{i=1}^n w_i(V, s_0)Y_i^l}{y_l(s_0) := sum_i(w_i(V, s_0) Y_i^l)}
//'
//' @rdname optStiefel
//' @keywords internal
double gradient(const arma::mat& X, double gradient(const arma::mat& X,
const arma::mat& X_diff, const arma::mat& X_diff,
const arma::mat& Y, const arma::mat& Y,
@ -178,9 +187,9 @@ double optStiefel(
double loss; double loss;
double error = datum::inf; double error = datum::inf;
double tau = tauInitial; double tau = tauInitial;
int count; int iter;
// main optimization loop // main optimization loop
for (count = 0; count < maxIter && error > tol; ++count) { for (iter = 0; iter < maxIter && error > tol; ++iter) {
// calc gradient `G = grad_V(L)(V)` // calc gradient `G = grad_V(L)(V)`
mat G; mat G;
loss = gradient(X, X_diff, Y, Y_rep, V, h, &G); loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
@ -191,8 +200,8 @@ double optStiefel(
// loss after step `L(V(tau))` // loss after step `L(V(tau))`
double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h); double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h);
// store `loss` in `history` and increase `count` // store `loss` in `history` and increase `iter`
history(count) = loss; history(iter) = loss;
// validate if loss decreased // validate if loss decreased
if ((loss_tau - loss) > slack * loss) { if ((loss_tau - loss) > slack * loss) {
@ -209,7 +218,7 @@ double optStiefel(
} }
// store final `loss` // store final `loss`
history(count) = loss; history(iter) = loss;
return loss; return loss;
} }
@ -233,7 +242,7 @@ double optStiefel(
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the //' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}. //' orthogonal space spaned by \code{V}.
//' //'
//' @rdname cve_cpp_V1 //' @rdname cve_cpp
//' @export //' @export
// [[Rcpp::export]] // [[Rcpp::export]]
Rcpp::List cve_cpp( Rcpp::List cve_cpp(

View File

@ -1,5 +1,5 @@
// //
// Standalone implementation for development. // Development file.
// //
// Usage: // Usage:
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V2.cpp')" // ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V2.cpp')"
@ -149,12 +149,12 @@ double optStiefel(
const int k, const int k,
const double h, const double h,
const double tauInitial, const double tauInitial,
const double rho1,
const double rho2,
const double tol, const double tol,
const int maxIter, const int maxIter,
const double rho1,
const double rho2,
const int maxLineSeachIter, const int maxLineSeachIter,
arma::mat& V, // out arma::mat& V, // out
arma::vec& history // out arma::vec& history // out
) { ) {
using namespace arma; using namespace arma;
@ -309,7 +309,7 @@ Rcpp::List cve_cpp(
mat V; // estimated `V` space mat V; // estimated `V` space
vec hist = vec(history.n_rows, fill::zeros); // optimization history vec hist = vec(history.n_rows, fill::zeros); // optimization history
double loss = optStiefel(X, Y, k, h, double loss = optStiefel(X, Y, k, h,
tauInitial, rho1, rho2, tol, maxIter, maxLineSeachIter, V, hist tauInitial, tol, maxIter, rho1, rho2, maxLineSeachIter, V, hist
); );
if (loss < loss_best) { if (loss < loss_best) {
loss_best = loss; loss_best = loss;

View File

@ -252,27 +252,49 @@ arma::mat grad_p(const arma::mat& X_ref,
loss /= n; loss /= n;
// scaling for gradient summation // scaling for gradient summation
for (uword k = 0; k < n; ++k) { // this scaling matrix is the lower triangular matrix defined as
for (uword l = 0; l < n; ++l) { //
// S_kl := (s_{kl} + s_{lk}) D_{kl}
// s_ij := (L_n(V, X_j) - (Y_i - y1(V, X_j))^2) W_ij
double factor;
for (uword l = 0; l < n; ++l) {
for (uword k = l + 1; k < n; ++k) {
tmp = Y[k] - y1[l]; tmp = Y[k] - y1[l];
S[l * n + k] = (L[l] - (tmp * tmp)) * W[l * n + k] * D[l * n + k]; factor = (L[l] - (tmp * tmp)) * W[l * n + k]; // \tile{S}_{kl}
tmp = Y[l] - y1[k];
factor += (L[k] - (tmp * tmp)) * W[k * n + l]; // \tile{S}_{lk}
S[l * n + k] = factor * D[l * n + k]; // (s_kl + s_lk) * D_kl
} }
} }
// gradient // gradient
double factor = -2. / (n * h * h); // reuse memory area of `Q`
for (uword j = 0; j < q; ++j) { // no longer needed and provides enough space (`q < p`)
double* GD = Q;
const double* X_l;
const double* X_k;
for (uword j = 0; j < p; ++j) {
for (uword i = 0; i < p; ++i) { for (uword i = 0; i < p; ++i) {
sum = 0.0; sum = 0.0;
for (uword k = 0; k < n; ++k) { for (uword l = 0; l < n; ++l) {
for (uword l = 0; l < n; ++l) { X_l = X + (l * p);
mvm = 0.0; for (uword k = l + 1; k < n; ++k) {
for (uword m = 0; m < p; ++m) { X_k = X + (k * p);
mvm += (X[l * p + m] - X[k * p + m]) * V[j * p + m]; sum += S[l * n + k] * (X_l[i] - X_k[i]) * (X_l[j] - X_k[j]);
}
sum += S[l * n + k] * (X[l * p + i] - X[k * p + i]) * mvm;
} }
} }
GD[j * p + i] = sum;
}
}
// distance gradient `DG` to gradient by multiplying with `V`
factor = -2. / (n * h * h);
for (uword i = 0; i < p; ++i) {
for (uword j = 0; j < q; ++j) {
sum = 0.0;
for (uword k = 0; k < p; ++k) {
sum += GD[k * p + i] * V[j * p + k];
}
G[j * p + i] = factor * sum; G[j * p + i] = factor * sum;
} }
} }
@ -352,16 +374,17 @@ setup.tests <- function () {
} }
counter <<- counter + 1 counter <<- counter + 1
} }
(mbm <- microbenchmark( mbm <- microbenchmark(
arma = arma_grad(X, X_diff, Y, Y_rep, V, h), arma = arma_grad(X, X_diff, Y, Y_rep, V, h),
grad = grad(Xt, Y, V, h), grad = grad(Xt, Y, V, h),
grad_p = grad_p(Xt, Y, V, h), grad_p = grad_p(Xt, Y, V, h),
check = comp.all, check = comp.all,
setup = setup.tests(), setup = setup.tests(),
times = 100L times = 100L
)) )
cat("\033[1m\033[92mTotal time:", format(Sys.time() - time.start), '\n')
cat("Total time:", format(Sys.time() - time.start), '\n') print(mbm)
cat("\033[0m")
boxplot(mbm, las = 2, xlab = NULL) boxplot(mbm, las = 2, xlab = NULL)