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wip: cleanup

This commit is contained in:
Daniel Kapla 2019-09-16 11:23:09 +02:00
parent 03153ef05e
commit 6f46cb2eaf
6 changed files with 5 additions and 874 deletions
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4
CVE_R/R/CVE.R
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@ -134,8 +134,8 @@ cve.call <- function(X, Y, method = "simple", nObs = nrow(X)^.5,
}
# augment result information
dr.k$method <- method
dr.k$call <- call
dr$method <- method
dr$call <- call
class(dr) <- "cve"
return(dr)
}

173
cve_V0.R
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@ -1,173 +0,0 @@
#' Euclidean vector norm (2-norm)
#'
#' @param x Numeric vector
#' @return Numeric
norm2 <- function(x) { return(sum(x^2)) }
#' Samples uniform from the Stiefel Manifold
#'
#' @param p row dim.
#' @param q col dim.
#' @return `(p, q)` semi-orthogonal matrix
rStiefl <- function(p, q) {
return(qr.Q(qr(matrix(rnorm(p * q, 0, 1), p, q))))
}
#' Matrix Trace
#'
#' @param M Square matrix
#' @return Trace \eqn{Tr(M)}
Tr <- function(M) {
return(sum(diag(M)))
}
#' Null space basis of given matrix `B`
#'
#' @param B `(p, q)` matrix
#' @return Semi-orthogonal `(p, p - q)` matrix `Q` spaning the null space of `B`
null <- function(M) {
tmp <- qr(M)
set <- if(tmp$rank == 0L) seq_len(ncol(M)) else -seq_len(tmp$rank)
return(qr.Q(tmp, complete = TRUE)[, set, drop = FALSE])
}
####
#chooses bandwith h according to formula in paper
#dim...dimension of X vector
#k... row dim of V (dim times q matrix) corresponding to a basis of orthogonal complement of B in model
# N...sample size
#nObs... nObs in bandwith formula
#tr...trace of sample covariance matrix of X
estimateBandwidth<-function(X, k, nObs) {
n <- nrow(X)
p <- ncol(X)
X_centered <- scale(X, center = TRUE, scale = FALSE)
Sigma <- (1 / n) * t(X_centered) %*% X_centered
quantil <- qchisq((nObs - 1) / (n - 1), k)
return(2 * quantil * Tr(Sigma) / p)
}
###########
# evaluates L(V) and returns L_n(V),(L_tilde_n(V,X_i))_{i=1,..,n} and grad_V L_n(V) (p times k)
# V... (dim times q) matrix
# Xl... output of Xl_fun
# dtemp...vector with pairwise distances |X_i - X_j|
# q...output of q_ind function
# Y... vector with N Y_i values
# if grad=T, gradient of L(V) also returned
LV <- function(V, Xl, dtemp, h, q, Y, grad = TRUE) {
N <- length(Y)
k <- if (is.vector(V)) { 1 } else { ncol(V) }
Xlv <- Xl %*% V
d <- dtemp - ((Xlv^2) %*% rep(1, k))
w <- dnorm(d / h) / dnorm(0)
w <- matrix(w, N, q)
w <- apply(w, 2, function(x) { x / sum(x) })
y1 <- t(w) %*% Y
y2 <- t(w) %*% (Y^2)
sig <- y2 - y1^2
result <- list(var = mean(sig), sig = sig)
if (grad == TRUE) {
tmp1 <- (kronecker(sig, rep(1, N)) - (as.vector(kronecker(rep(1, q), Y)) - kronecker(y1, rep(1, N)))^2)
if (k == 1) {
grad_d <- -2 * Xl * as.vector(Xlv)
grad <- (1 / h^2) * (1 / q) * t(grad_d * as.vector(d) * as.vector(w)) %*% tmp1
} else {
grad <- matrix(0, nrow(V), ncol(V))
for (j in 1:k) {
grad_d <- -2 * Xl * as.vector(Xlv[ ,j])
grad[ ,j] <- (1 / h^2) * (1 / q) * t(grad_d * as.vector(d) * as.vector(w)) %*% tmp1
}
}
result$grad = grad
}
return(result)
}
#### performs stiefle optimization of argmin_{V : V'V=I_k} L_n(V)
#through curvilinear search with k0 starting values drawn uniformly on stiefel maniquefold
#dat...(N times dim+1) matrix with first column corresponding to Y values, the other columns
#consists of X data matrix, (i.e. dat=cbind(Y,X))
#h... bandwidth
#k...row dimension of V that is calculated, corresponds to dimension of orthogonal complement of B
#k0... number of arbitrary starting values
#p...fraction of data points used as shifting point
#maxIter... number of maximal iterations in curvilinear search
#nObs.. nObs parameter for choosing bandwidth if no h is supplied
#lambda_0...initial stepsize
#tol...tolerance for stoping iterations
#sclack_para...if relative improvment is worse than sclack_para the stepsize is reduced
#output:
#est_base...Vhat_k= argmin_V:V'V=I_k L_n(V) a (dim times k) matrix where dim is row-dimension of X data matrix
#var...value of L_n(Vhat_k)
#aov_dat... (L_tilde_n(Vhat_k,X_i))_{i=1,..,N}
#count...number of iterations
#h...bandwidth
cve_R <- function(
X, Y, k,
nObs = sqrt(nrow(X)),
tauInitial = 1.0,
tol = 1e-3,
slack = 0,
maxIter = 50L,
attempts = 10L
) {
# get dimensions
n <- nrow(X)
p <- ncol(X)
q <- p - k
Xl <- kronecker(rep(1, n), X) - kronecker(X, rep(1, n))
Xd <- apply(Xl, 1, norm2)
I_p <- diag(1, p)
# estimate bandwidth
h <- estimateBandwidth(X, k, nObs)
Lbest <- Inf
Vend <- mat.or.vec(p, q)
for (. in 1:attempts) {
Vnew <- Vold <- rStiefl(p, q)
Lnew <- Lold <- exp(10000)
tau <- tauInitial
error <- Inf
count <- 0
while (error > tol & count < maxIter) {
tmp <- LV(Vold, Xl, Xd, h, n, Y)
G <- tmp$grad
Lold <- tmp$var
W <- tau * (G %*% t(Vold) - Vold %*% t(G))
Vnew <- solve(I_p + W) %*% (I_p - W) %*% Vold
Lnew <- LV(Vnew, Xl, Xd, h, n, Y, grad = FALSE)$var
if ((Lnew - Lold) > slack * Lold) {
tau = tau / 2
error <- Inf
} else {
error <- norm(Vold %*% t(Vold) - Vnew %*% t(Vnew), "F") / sqrt(2 * k)
Vold <- Vnew
}
count <- count + 1
}
if (Lbest > Lnew) {
Lbest <- Lnew
Vend <- Vnew
}
}
return(list(
loss = Lbest,
V = Vend,
B = null(Vend),
h = h
))
}

330
cve_V1.cpp
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@ -1,330 +0,0 @@
//
// Development file.
//
// Usage:
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')"
//
// only `RcppArmadillo.h` which includes `Rcpp.h`
#include <RcppArmadillo.h>
// through the depends attribute `Rcpp` is tolled to create
// hooks for `RcppArmadillo` needed by the build process.
//
// [[Rcpp::depends(RcppArmadillo)]]
// required for `R::qchisq()` used in `estimateBandwidth()`
#include <Rmath.h>
//' Estimated bandwidth for CVE.
//'
//' Estimates a propper bandwidth \code{h} according
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
//'
//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
//' q. Therefor each row represents a datapoint of dimension p.
//' @param k Guess for rank(B).
//' @param nObs Ether numeric of a function. If specified as numeric value
//' its used in the computation of the bandwidth directly. If its a function
//' `nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
//' supplied at all is to use \code{nObs <- nrow(x)^0.5}.
//'
//' @seealso [qchisq()]
//'
//' @export
// [[Rcpp::export]]
double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
using namespace arma;
uword n = X.n_rows; // nr samples
uword p = X.n_cols; // dimension of rand. var. `X`
// column mean
mat M = mean(X);
// center `X`
mat C = X.each_row() - M;
// trace of covariance matrix, `traceSigma = Tr(C' C)`
double traceSigma = accu(C % C);
// compute estimated bandwidth
double qchi2 = R::qchisq((nObs - 1.0) / (n - 1), static_cast<double>(k), 1, 0);
return 2.0 * qchi2 * traceSigma / (p * n);
}
//' Random element from Stiefel Manifold `S(p, q)`.
//'
//' Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
//' This is done by taking the Q-component of the QR decomposition
//' from a `(p, q)` Matrix with independent standart normal entries.
//' As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
//'
//' @param p Row dimension
//' @param q Column dimension
//'
//' @return Matrix of dim `(p, q)`.
//'
//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
//'
//' @export
// [[Rcpp::export]]
arma::mat rStiefel(arma::uword p, arma::uword q) {
arma::mat Q, R;
arma::qr_econ(Q, R, arma::randn<arma::mat>(p, 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,
const arma::mat& X_diff,
const arma::mat& Y,
const arma::mat& Y_rep,
const arma::mat& V,
const double h,
arma::mat* G = 0
) {
using namespace arma;
uword n = X.n_rows;
uword p = X.n_cols;
// orthogonal projection matrix `Q = I - VV'` for dist computation
mat Q = -(V * V.t());
Q.diag() += 1;
// calc pairwise distances as `D` with `D(i, j) = d_i(V, X_j)`
vec D_vec = sum(square(X_diff * Q), 1);
mat D = reshape(D_vec, n, n);
// calc weights as `W` with `W(i, j) = w_i(V, X_j)`
mat W = exp(D / (-2.0 * h));
// column-wise normalization via 1-norm
W = normalise(W, 1);
vec W_vec = vectorise(W);
// weighted `Y` means (first and second order)
vec y1 = W.t() * Y;
vec y2 = W.t() * square(Y);
// loss for each `X_i`, meaning `L(i) = L_n(V, X_i)`
vec L = y2 - square(y1);
// `loss = L_n(V)`
double loss = mean(L);
// check if gradient as output variable is set
if (G != 0) {
// `G = grad(L_n(V))` a.k.a. gradient of `L` with respect to `V`
vec scale = (repelem(L, n, 1) - square(Y_rep - repelem(y1, n, 1))) % W_vec % D_vec;
mat X_diff_scale = X_diff.each_col() % scale;
(*G) = X_diff_scale.t() * X_diff * V;
(*G) *= -2.0 / (h * h * n);
}
return loss;
}
//' Stiefel Optimization.
//'
//' 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}
//' to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
//' span(B) = orth(span(B))}.
//'
//' @param X data points
//' @param Y response
//' @param k assumed \eqn{rank(B)}
//' @param nObs parameter for bandwidth estimation, typical value
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
//' @param tau Initial step size
//' @param tol Tolerance for update error used for stopping criterion
//' \eqn{|| V(j) V(j)' - V(j+1) V(j+1)' ||_2 < tol}{%
//' \| V_j V_j' - V_{j+1} V_{j+1}' \|_2 < tol}.
//' @param maxIter Upper bound of optimization iterations
//'
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}.
//'
//' @rdname optStiefel
//' @keywords internal
double optStiefel(
const arma::mat& X,
const arma::vec& Y,
const int k,
const double h,
const double tauInitial,
const double tol,
const double slack,
const int maxIter,
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);
// initial start value for `V`
V = rStiefel(p, q);
// init optimization `loss`es, `error` and stepsize `tau`
// double loss_next = datum::inf;
double loss;
double error = datum::inf;
double tau = tauInitial;
int iter;
// main optimization loop
for (iter = 0; iter < maxIter && error > tol; ++iter) {
// calc gradient `G = grad_V(L)(V)`
mat G;
loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
// matrix `A` for colescy-transform of the gradient
mat A = tau * ((G * V.t()) - (V * G.t()));
// next iteration step of `V`
mat V_tau = inv(I_p + A) * (I_p - A) * V;
// loss after step `L(V(tau))`
double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h);
// store `loss` in `history` and increase `iter`
history(iter) = loss;
// validate if loss decreased
if ((loss_tau - loss) > slack * loss) {
// ignore step, retry with half the step size
tau = tau / 2.;
error = datum::inf;
} else {
// compute step error (break condition)
error = norm((V * V.t()) - (V_tau * V_tau.t()), 2) / (2 * q);
// shift for next iteration
V = V_tau;
loss = loss_tau;
}
}
// store final `loss`
history(iter) = loss;
return loss;
}
//' Conditional Variance Estimation (CVE) method.
//'
//' This version uses a "simple" stiefel optimization schema.
//'
//' @param X data points
//' @param Y response
//' @param k assumed \eqn{rank(B)}
//' @param nObs parameter for bandwidth estimation, typical value
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
//' @param tau Initial step size (default 1)
//' @param tol Tolerance for update error used for stopping criterion (default 1e-5)
//' @param slack Ratio of small negative error allowed in loss optimization (default -1e-10)
//' @param maxIter Upper bound of optimization iterations (default 50)
//' @param attempts Number of tryes with new random optimization starting points (default 10)
//'
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}.
//'
//' @rdname cve_cpp
//' @export
// [[Rcpp::export]]
Rcpp::List cve_cpp(
const arma::mat& X,
const arma::vec& Y,
const int k,
const double nObs,
const double tauInitial = 1.,
const double tol = 1e-5,
const double slack = -1e-10,
const int maxIter = 50,
const int attempts = 10
) {
using namespace arma;
// tracker of current best results
mat V_best;
double loss_best = datum::inf;
// estimate bandwidth
double h = estimateBandwidth(X, k, nObs);
// loss history for each optimization attempt
// each column contaions the iteration history for the loss
mat history = mat(maxIter + 1, attempts);
// multiple stiefel optimization attempts
for (int i = 0; i < attempts; ++i) {
// declare output variables
mat V; // estimated `V` space
vec hist = vec(history.n_rows, fill::zeros); // optimization history
double loss = optStiefel(X, Y, k, h, tauInitial, tol, slack, maxIter, V, hist);
if (loss < loss_best) {
loss_best = loss;
V_best = V;
}
// write history to history collection
history.col(i) = hist;
}
// get `B` as kernal of `V.t()`
mat B = null(V_best.t());
return Rcpp::List::create(
Rcpp::Named("history") = history,
Rcpp::Named("loss") = loss_best,
Rcpp::Named("h") = h,
Rcpp::Named("V") = V_best,
Rcpp::Named("B") = B
);
}
/*** R
source("CVE/R/datasets.R")
ds <- dataset()
print(system.time(
cve.res <- cve_cpp(
X = ds$X,
Y = ds$Y,
k = ncol(ds$B),
nObs = sqrt(nrow(ds$X))
)
))
pdf('plots/cve_V1_history.pdf')
H <- cve.res$history
H_i <- H[H[, 1] > 0, 1]
plot(1:length(H_i), H_i,
main = "History cve_V1",
xlab = "Iterations i",
ylab = expression(loss == L[n](V^{(i)})),
xlim = c(1, nrow(H)),
ylim = c(0, max(H)),
type = "l"
)
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)
*/

367
cve_V2.cpp
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@ -1,367 +0,0 @@
//
// Development file.
//
// Usage:
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V2.cpp')"
//
// only `RcppArmadillo.h` which includes `Rcpp.h`
#include <RcppArmadillo.h>
// through the depends attribute `Rcpp` is tolled to create
// hooks for `RcppArmadillo` needed by the build process.
//
// [[Rcpp::depends(RcppArmadillo)]]
// required for `R::qchisq()` used in `estimateBandwidth()`
#include <Rmath.h>
//' Estimated bandwidth for CVE.
//'
//' Estimates a propper bandwidth \code{h} according
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
//'
//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
//' q. Therefor each row represents a datapoint of dimension p.
//' @param k Guess for rank(B).
//' @param nObs Ether numeric of a function. If specified as numeric value
//' its used in the computation of the bandwidth directly. If its a function
//' `nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
//' supplied at all is to use \code{nObs <- nrow(x)^0.5}.
//'
//' @seealso [qchisq()]
//'
//' @export
// [[Rcpp::export]]
double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
using namespace arma;
uword n = X.n_rows; // nr samples
uword p = X.n_cols; // dimension of rand. var. `X`
// column mean
mat M = mean(X);
// center `X`
mat C = X.each_row() - M;
// trace of covariance matrix, `traceSigma = Tr(C' C)`
double traceSigma = accu(C % C);
// compute estimated bandwidth
double qchi2 = R::qchisq((nObs - 1.0) / (n - 1), static_cast<double>(k), 1, 0);
return 2.0 * qchi2 * traceSigma / (p * n);
}
//' Random element from Stiefel Manifold `S(p, q)`.
//'
//' Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
//' This is done by taking the Q-component of the QR decomposition
//' from a `(p, q)` Matrix with independent standart normal entries.
//' As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
//'
//' @param p Row dimension
//' @param q Column dimension
//'
//' @return Matrix of dim `(p, q)`.
//'
//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
//'
//' @export
// [[Rcpp::export]]
arma::mat rStiefel(arma::uword p, arma::uword q) {
arma::mat Q, R;
arma::qr_econ(Q, R, arma::randn<arma::mat>(p, q));
return Q;
}
double gradient(const arma::mat& X,
const arma::mat& X_diff,
const arma::mat& Y,
const arma::mat& Y_rep,
const arma::mat& V,
const double h,
arma::mat* G = 0
) {
using namespace arma;
uword n = X.n_rows;
uword p = X.n_cols;
// orthogonal projection matrix `Q = I - VV'` for dist computation
mat Q = -(V * V.t());
Q.diag() += 1;
// calc pairwise distances as `D` with `D(i, j) = d_i(V, X_j)`
vec D_vec = sum(square(X_diff * Q), 1);
mat D = reshape(D_vec, n, n);
// calc weights as `W` with `W(i, j) = w_i(V, X_j)`
mat W = exp(D / (-2.0 * h));
// column-wise normalization via 1-norm
W = normalise(W, 1);
vec W_vec = vectorise(W);
// weighted `Y` means (first and second order)
vec y1 = W.t() * Y;
vec y2 = W.t() * square(Y);
// loss for each `X_i`, meaning `L(i) = L_n(V, X_i)`
vec L = y2 - square(y1);
// `loss = L_n(V)`
double loss = mean(L);
// check if gradient as output variable is set
if (G != 0) {
// `G = grad(L_n(V))` a.k.a. gradient of `L` with respect to `V`
vec scale = (repelem(L, n, 1) - square(Y_rep - repelem(y1, n, 1))) % W_vec % D_vec;
mat X_diff_scale = X_diff.each_col() % scale;
(*G) = X_diff_scale.t() * X_diff * V;
(*G) *= -2.0 / (h * h * n);
}
return loss;
}
//' Stiefel Optimization with curvilinear linesearch.
//'
//' TODO: finish doc. comment
//' 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}
//' to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
//' span(B) = orth(span(B))}.
//' The curvilinear linesearch uses Armijo-Wolfe conditions:
// \deqn{L(V(tau)) > L(V(0)) + rho_1 * tau * L(V(0))'}
//' \deqn{L(V(tau))' < rho_2 * L(V(0))'}
//'
//' @param X data points
//' @param Y response
//' @param k assumed \eqn{rank(B)}
//' @param nObs parameter for bandwidth estimation, typical value
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
//' @param tau Initial step size
//' @param tol Tolerance for update error used for stopping criterion
//' @param maxIter Upper bound of optimization iterations
//'
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}.
//'
//' @rdname optStiefel
//' @keywords internal
double optStiefel(
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 maxLineSeachIter,
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;
// 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 < maxLineSeachIter);
// 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;
}
//' Conditional Variance Estimation (CVE) method.
//'
//' This version uses a curvilinear linesearch for the stiefel optimization.
//'
//' @param X data points
//' @param Y response
//' @param k assumed \eqn{rank(B)}
//' @param nObs parameter for bandwidth estimation, typical value
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
//' @param tau Initial step size (default 1)
//' @param tol Tolerance for update error used for stopping criterion (default 1e-5)
//' @param slack Ratio of small negative error allowed in loss optimization (default -1e-10)
//' @param maxIter Upper bound of optimization iterations (default 50)
//' @param attempts Number of tryes with new random optimization starting points (default 10)
//'
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
//' orthogonal space spaned by \code{V}.
//'
//' @rdname cve_cpp_V2
//' @export
// [[Rcpp::export]]
Rcpp::List cve_cpp(
const arma::mat& X,
const arma::vec& Y,
const int k,
const double nObs,
const double tauInitial = 1.,
const double rho1 = 0.05,
const double rho2 = 0.95,
const double tol = 1e-6,
const int maxIter = 50,
const int maxLineSeachIter = 10,
const int attempts = 10
) {
using namespace arma;
// tracker of current best results
mat V_best;
double loss_best = datum::inf;
// estimate bandwidth
double h = estimateBandwidth(X, k, nObs);
// loss history for each optimization attempt
// each column contaions the iteration history for the loss
mat history = mat(maxIter + 1, attempts);
// multiple stiefel optimization attempts
for (int i = 0; i < attempts; ++i) {
// declare output variables
mat V; // estimated `V` space
vec hist = vec(history.n_rows, fill::zeros); // optimization history
double loss = optStiefel(X, Y, k, h,
tauInitial, tol, maxIter, rho1, rho2, maxLineSeachIter, V, hist
);
if (loss < loss_best) {
loss_best = loss;
V_best = V;
}
// write history to history collection
history.col(i) = hist;
}
// get `B` as kernal of `V.t()`
mat B = null(V_best.t());
return Rcpp::List::create(
Rcpp::Named("history") = history,
Rcpp::Named("loss") = loss_best,
Rcpp::Named("h") = h,
Rcpp::Named("V") = V_best,
Rcpp::Named("B") = B
);
}
/*** R
source("CVE/R/datasets.R")
ds <- dataset()
print(system.time(
cve.res <- cve_cpp(
X = ds$X,
Y = ds$Y,
k = ncol(ds$B),
nObs = sqrt(nrow(ds$X))
)
))
pdf('plots/cve_V2_history.pdf')
H <- cve.res$history
H_i <- H[H[, 1] > 0, 1]
plot(1:length(H_i), H_i,
main = "History cve_V2",
xlab = "Iterations i",
ylab = expression(loss == L[n](V^{(i)})),
xlim = c(1, nrow(H)),
ylim = c(0, max(H)),
type = "l"
)
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)
*/

3
runtime_test.R
View File

@ -1,5 +1,7 @@
# Usage:
# ~$ Rscript runtime_test.R
# library(CVEpureR) # load CVE's pure R implementation
library(CVE) # load CVE
#' Writes log information to console. (to not get bored^^)
tell.user <- function(name, start.time, i, length) {
@ -24,7 +26,6 @@ dataset.names <- c("M1", "M2", "M3", "M4", "M5")
# Set used CVE method
methods <- c("simple") # c("legacy", "simple", "sgd", "linesearch")
library(CVE) # load CVE
if ("legacy" %in% methods) {
# Source legacy code (but only if needed)
source("CVE_legacy/function_script.R")