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@ -289,7 +289,7 @@ cve.call <- function(X, Y, method = "simple",
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} else {
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} else {
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tol <- as.double(tol)
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tol <- as.double(tol)
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}
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}
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if (!is.numeric(slack) || length(slack) > 1L || slack < 0.0) {
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if (!is.numeric(slack) || length(slack) > 1L) {
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stop("Break condition slack 'slack' must be not negative number.")
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stop("Break condition slack 'slack' must be not negative number.")
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} else {
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} else {
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slack <- as.double(slack)
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slack <- as.double(slack)
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@ -1,3 +1,83 @@
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#'
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#' @param n number of samples.
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#' @param mu mean
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#' @param sigma covariance matrix.
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#'
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#' @returns a \eqn{n\times p} matrix with samples in its rows.
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#'
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#' @examples
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#' rmvnorm(20, sigma = matrix(c(2, 1, 1, 2), 2))
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#' rmvnorm(20, mu = c(3, -1, 2))
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rmvnorm <- function(n = 1, mu = rep(0, p), sigma = diag(p)) {
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if (!missing(sigma)) {
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p <- nrow(sigma)
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} else if (!missing(mu)) {
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mu <- matrix(mu, ncol = 1)
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p <- nrow(mu)
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} else {
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stop("At least one of 'mu' or 'sigma' must be supplied.")
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}
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# See: https://en.wikipedia.org/wiki/Multivariate_normal_distribution
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return(rep(mu, each = n) + matrix(rnorm(n * p), n) %*% chol(sigma))
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}
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#' Samples from the multivariate t distribution (student distribution).
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#'
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#' @param n number of samples.
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#' @param mu mean, ... TODO:
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#' @param sigma a \eqn{k\times k} positive definite matrix. If the degree
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#' \eqn{\nu} if bigger than 2 the created covariance is
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#' \deqn{var(x) = \Sigma\frac{\nu}{\nu - 2}}
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#' for \eqn{\nu > 2}.
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#' @param df degree of freedom \eqn{\nu}.
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#'
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#' @returns a \eqn{n\times p} matrix with samples in its rows.
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#'
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#' @examples
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#' rmvt(20, c(0, 1), matrix(c(3, 1, 1, 2), 2), 3)
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#' rmvt(20, sigma = matrix(c(2, 1, 1, 2), 2), 3)
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#' rmvt(20, mu = c(3, -1, 2), 3)
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rmvt <- function(n = 1, mu = rep(0, p), sigma = diag(p), df = Inf) {
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if (!missing(sigma)) {
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p <- nrow(sigma)
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} else if (!missing(mu)) {
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mu <- matrix(mu, ncol = 1)
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p <- nrow(mu)
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} else {
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stop("At least one of 'mu' or 'sigma' must be supplied.")
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}
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if (df == Inf) {
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Z <- 1
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} else {
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Z <- sqrt(df / rchisq(n, df))
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}
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return(rmvnorm(n, sigma = sigma) * Z + rep(mu, each = n))
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}
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#' Generalized Normal Distribution.
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#' see: https://en.wikipedia.org/wiki/Generalized_normal_distribution
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rgnorm <- function(n = 1, mu = 0, alpha = 1, beta = 1) {
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if (alpha <= 0 | beta <= 0) {
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stop("alpha and beta must be positive.")
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}
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lambda <- (1 / alpha)^beta
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scales <- qgamma(runif(n), shape = 1 / beta, scale = 1 / lambda)^(1 / beta)
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return(scales * ((-1)^rbinom(n, 1, 0.5)) + mu)
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}
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#' Laplace distribution
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#' see: https://en.wikipedia.org/wiki/Laplace_distribution
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rlaplace <- function(n = 1, mu = 0, sigma = 1) {
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U <- runif(n, -0.5, 0.5)
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scale <- sigma / sqrt(2)
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return(mu - scale * sign(U) * log(1 - 2 * abs(U)))
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}
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#' Generates test datasets.
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#' Generates test datasets.
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#'
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#'
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#' Provides sample datasets. There are 5 different datasets named
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#' Provides sample datasets. There are 5 different datasets named
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@ -41,72 +121,73 @@
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#' @import stats
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#' @import stats
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#' @importFrom stats rnorm rbinom
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#' @importFrom stats rnorm rbinom
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#' @export
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#' @export
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dataset <- function(name = "M1", n, B, p.mix = 0.3, lambda = 1.0) {
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dataset <- function(name = "M1", n = NULL, p = 20, sigma = 0.5, ...) {
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# validate parameters
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name <- toupper(name)
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stopifnot(name %in% c("M1", "M2", "M3", "M4", "M5"))
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if (nchar(name) == 1) { name <- paste0("M", name) }
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# set default values if not supplied
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if (missing(n)) {
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n <- if (name %in% c("M1", "M2")) 200 else if (name != "M5") 100 else 42
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}
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if (missing(B)) {
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p <- 12
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if (name == "M1") {
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B <- cbind(
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c( 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0),
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c( 1,-1, 1,-1, 1,-1, 0, 0, 0, 0, 0, 0)
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) / sqrt(6)
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} else if (name == "M2") {
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B <- cbind(
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c(c(1, 0), rep(0, 10)),
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c(c(0, 1), rep(0, 10))
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)
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} else {
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B <- matrix(c(rep(1 / sqrt(6), 6), rep(0, 6)), 12, 1)
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}
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} else {
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p <- nrow(B)
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# validate col. nr to match dataset `k = ncol(B)`
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stopifnot(
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name %in% c("M1", "M2") && ncol(B) == 2,
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name %in% c("M3", "M4", "M5") && ncol(B) == 1
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)
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}
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# set link function `g` for model `Y ~ g(B'X) + epsilon`
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if (name == "M1") {
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if (name == "M1") {
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g <- function(BX) { BX[1] / (0.5 + (BX[2] + 1.5)^2) }
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if (missing(n)) { n <- 100 }
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# B ... `p x 1`
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B <- matrix(c(rep(1 / sqrt(6), 6), rep(0, p - 6)), ncol = 1)
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X <- rmvnorm(n, sigma = sigma^abs(outer(1:p, 1:p, FUN = `-`)))
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beta <- 0.5
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Y <- cos(X %*% B) + rgnorm(n, 0,
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alpha = sqrt(0.25 * gamma(1 / beta) / gamma(3 / beta)),
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beta = beta
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)
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} else if (name == "M2") {
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} else if (name == "M2") {
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g <- function(BX) { BX[1] * BX[2]^2 }
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if (missing(n)) { n <- 100 }
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} else if (name %in% c("M3", "M4")) {
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prob <- 0.3
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g <- function(BX) { cos(BX[1]) }
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lambda <- 1 # dispersion
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} else { # name == "M5"
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# B ... `p x 1`
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g <- function(BX) { 2 * log(abs(BX[1]) + 1) }
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B <- matrix(c(rep(1 / sqrt(6), 6), rep(0, p - 6)), ncol = 1)
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Z <- 2 * rbinom(n, 1, prob) - 1
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X <- matrix(rep(lambda * Z, p) + rnorm(n * p), n)
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Y <- cos(X %*% B) + rnorm(n, 0, sigma)
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} else if (name == "M3") {
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if (missing(n)) { n <- 200 }
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# B ... `p x 1`
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B <- matrix(c(rep(1 / sqrt(6), 6), rep(0, p - 6)), ncol = 1)
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X <- matrix(rnorm(n * p), n)
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Y <- 1.5 * log(2 + abs(X %*% B)) + rnorm(n, 0, sigma^2)
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} else if (name == "M4") {
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if (missing(n)) { n <- 200 }
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# B ... `p x 2`
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B <- cbind(
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c(rep(1 / sqrt(6), 6), rep(0, p - 6)),
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c(rep(c(1, -1), 3) / sqrt(6), rep(0, p - 6))
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)
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X <- rmvnorm(n, sigma = sigma^abs(outer(1:p, 1:p, FUN = `-`)))
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XB <- X %*% B
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Y <- (XB[, 1]) / (0.5 + (XB[, 2] + 1.5)^2) + rnorm(n, 0, sigma^2)
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} else if (name == "M5") {
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if (missing(n)) { n <- 200 }
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# B ... `p x 2`
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B <- cbind(
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c(rep(1, 6), rep(0, p - 6)),
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c(rep(c(1, -1), 3), rep(0, p - 6))
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) / sqrt(6)
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X <- matrix(runif(n * p), n)
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XB <- X %*% B
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Y <- cos(XB[, 1] * pi) * (XB[, 2] + 1)^2 + rnorm(n, 0, sigma^2)
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} else if (name == "M6") {
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if (missing(n)) { n <- 200 }
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# B ... `p x 3`
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B <- diag(p)[, -(3:(p - 1))]
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X <- matrix(rnorm(n * p), n)
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Y <- rowSums((X %*% B)^2) + rnorm(n, 0, sigma^2)
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} else if (name == "M7") {
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if (missing(n)) { n <- 400 }
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# B ... `p x 4`
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B <- diag(p)[, -(4:(p - 1))]
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# "R"andom "M"ulti"V"ariate "S"tudent
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X <- rmvt(n = n, sigma = diag(p), df = 3)
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XB <- X %*% B
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Y <- (XB[, 1]) * (XB[, 2])^2 + (XB[, 3]) * (XB[, 4])
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Y <- Y + rlaplace(n, 0, sigma)
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} else {
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stop("Got unknown dataset name.")
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}
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}
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# compute X
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if (name != "M4") {
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# compute root of the covariance matrix according the dataset
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if (name %in% c("M1", "M3")) {
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# Variance-Covariance structure for `X ~ N_p(0, \Sigma)` with
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# `\Sigma_{i, j} = 0.5^{|i - j|}`.
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Sigma <- matrix(0.5^abs(kronecker(1:p, 1:p, '-')), p, p)
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# decompose Sigma to Sigma.root^T Sigma.root = Sigma for usage in creation of `X`
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Sigma.root <- chol(Sigma)
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} else { # name %in% c("M2", "M5")
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Sigma.root <- diag(rep(1, p)) # d-dim identity
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}
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# data `X` as multivariate random normal variable with
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# variance matrix `Sigma`.
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X <- replicate(p, rnorm(n, 0, 1)) %*% Sigma.root
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} else { # name == "M4"
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X <- t(replicate(100, rep((1 - 2 * rbinom(1, 1, p.mix)) * lambda, p) + rnorm(p, 0, 1)))
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}
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# responce `y ~ g(B'X) + epsilon` with `epsilon ~ N(0, 1 / 2)`
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Y <- apply(X, 1, function(X_i) {
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g(t(B) %*% X_i) + rnorm(1, 0, 0.5)
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})
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return(list(X = X, Y = Y, B = B, name = name))
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return(list(X = X, Y = Y, B = B, name = name))
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}
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}
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#' @param X data matrix with samples in its rows.
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#' @param X data matrix with samples in its rows.
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#' @param k Dimension of lower dimensional projection.
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#' @param k Dimension of lower dimensional projection.
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#' @param nObs number of points in a slice, see \eqn{nObs} in CVE paper.
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#' @param nObs number of points in a slice, see \eqn{nObs} in CVE paper.
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#' @param version either \code{1} or \code{2}, where
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#' \itemize{
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#' \item 1: uses the following formula:
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#' \deqn{%
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#' h = (2 * tr(\Sigma) / p) * (1.2 * n^{-1 / (4 + k)})^2}{%
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#' h = (2 * tr(\Sigma) / p) * (1.2 * n^(\frac{-1}{4 + k}))^2}
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#' \item 2: uses
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#' \deqn{%
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#' h = (2 * tr(\Sigma) / p) * \chi_k^-1((nObs - 1) / (n - 1))}{%
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#' h = (2 * tr(\Sigma) / p) * \chi_k^{-1}(\frac{nObs - 1}{n - 1})}
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#' }
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#'
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#'
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#' @return Estimated bandwidth \code{h}.
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#' @return Estimated bandwidth \code{h}.
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#'
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#'
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#' print(cve.obj.simple$res$'1'$h)
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#' print(cve.obj.simple$res$'1'$h)
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#' print(estimate.bandwidth(x, k = k))
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#' print(estimate.bandwidth(x, k = k))
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#' @export
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#' @export
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estimate.bandwidth <- function(X, k, nObs) {
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estimate.bandwidth <- function (X, k, nObs, version = 1L) {
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n <- nrow(X)
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n <- nrow(X)
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p <- ncol(X)
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p <- ncol(X)
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if (version == 1) {
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X_centered <- scale(X, center = TRUE, scale = FALSE)
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X_centered <- scale(X, center = TRUE, scale = FALSE)
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Sigma <- crossprod(X_centered, X_centered) / n
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Sigma <- crossprod(X_centered, X_centered)/n
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return((2 * sum(diag(Sigma))/p) * (1.2 * n^(-1/(4 + k)))^2)
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return((2 * sum(diag(Sigma)) / p) * (1.2 * n^(-1 / (4 + k)))^2)
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} else if (version == 2) {
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X_c <- scale(X, center = TRUE, scale = FALSE)
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return(2 * qchisq((nObs - 1) / (n - 1), k) * sum(X_c^2) / (n * p))
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} else {
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stop("Unknown version.")
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}
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}
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}
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#' Predicts SDR dimension using \code{\link[mda]{mars}} via a Cross-Validation.
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predict_dim_cv <- function(object) {
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#'
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#' @param object instance of class \code{cve} (result of \code{cve},
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#' \code{cve.call}).
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#' @param ... ignored.
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#'
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#' @return list with
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#' \itemize{
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#' \item MSE: Mean Square Error,
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#' \item k: predicted dimensions.
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#' }
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#'
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#' @examples
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#' # create B for simulation
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#' B <- rep(1, 5) / sqrt(5)
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#'
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#' set.seed(21)
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#' # creat predictor data x ~ N(0, I_p)
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#' x <- matrix(rnorm(500), 100)
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#'
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#' # simulate response variable
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#' # y = f(B'x) + err
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#' # with f(x1) = x1 and err ~ N(0, 0.25^2)
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#' y <- x %*% B + 0.25 * rnorm(100)
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#'
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#' # Calculate cve for unknown k between min.dim and max.dim.
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#' cve.obj.simple <- cve(y ~ x)
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#'
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#' predict_dim(cve.obj.simple)
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#'
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#' @export
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predict_dim <- function(object, ...) {
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# Get centered training data and dimensions
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# Get centered training data and dimensions
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X <- scale(object$X, center = TRUE, scale = FALSE)
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X <- scale(object$X, center = TRUE, scale = FALSE)
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n <- nrow(object$X) # umber of training data samples
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n <- nrow(object$X) # umber of training data samples
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k = as.integer(names(which.min(MSE)))
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k = as.integer(names(which.min(MSE)))
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))
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))
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}
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}
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# TODO: write doc
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predict_dim_elbow <- function(object) {
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# extract original data from object (cve result)
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X <- object$X
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Y <- object$Y
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# Get dimensions
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n <- nrow(X)
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p <- ncol(X)
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# Compute persistent data.
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i = rep(1:n, n)
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j = rep(1:n, each = n)
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D.eucl = matrix((X[i, ] - X[j, ])^2 %*% rep(1, p), n)
|
||||||
|
|
||||||
|
losses <- vector("double", length(object$res))
|
||||||
|
names(losses) <- names(object$res)
|
||||||
|
# Compute per sample losses with alternative bandwidth for each dimension.
|
||||||
|
for (dr.k in object$res) {
|
||||||
|
# extract dimension specific estimates and dimensions.
|
||||||
|
k <- dr.k$k
|
||||||
|
V <- dr.k$V
|
||||||
|
q <- ncol(V)
|
||||||
|
# estimate bandwidth according alternative formula (see: TODO: see)
|
||||||
|
h <- estimate.bandwidth(X, k, sqrt(n), version = 2L)
|
||||||
|
# Projected `X`
|
||||||
|
XV <- X %*% V
|
||||||
|
# Devectorized distance matrix
|
||||||
|
# (inefficient in R but fast in C)
|
||||||
|
D <- matrix((XV[i, , drop = F] - XV[j, , drop = F])^2 %*% rep(1, q), n)
|
||||||
|
D <- D.eucl - D
|
||||||
|
# Apply kernel
|
||||||
|
K <- exp((-0.5 / h^2) * D^2)
|
||||||
|
# sum columns
|
||||||
|
colSumsK <- colSums(K)
|
||||||
|
# compute weighted and square meighted reponses
|
||||||
|
y1 <- (K %*% Y) / colSumsK
|
||||||
|
y2 <- (K %*% Y^2) / colSumsK
|
||||||
|
# total loss
|
||||||
|
losses[[as.character(k)]] <- mean(y2 - y1^2)
|
||||||
|
}
|
||||||
|
|
||||||
|
return(list(
|
||||||
|
losses = losses,
|
||||||
|
k = as.integer(names(which.min(losses)))
|
||||||
|
))
|
||||||
|
}
|
||||||
|
|
||||||
|
predict_dim_wilcoxon <- function(object, p.value = 0.05) {
|
||||||
|
# extract original data from object (cve result)
|
||||||
|
X <- object$X
|
||||||
|
Y <- object$Y
|
||||||
|
# Get dimensions
|
||||||
|
n <- nrow(X)
|
||||||
|
p <- ncol(X)
|
||||||
|
# Compute persistent data.
|
||||||
|
i = rep(1:n, n)
|
||||||
|
j = rep(1:n, each = n)
|
||||||
|
D.eucl = matrix((X[i, ] - X[j, ])^2 %*% rep(1, p), n)
|
||||||
|
|
||||||
|
L <- matrix(NA, n, length(object$res))
|
||||||
|
colnames(L) <- names(object$res)
|
||||||
|
# Compute per sample losses with alternative bandwidth for each dimension.
|
||||||
|
for (dr.k in object$res) {
|
||||||
|
# extract dimension specific estimates and dimensions.
|
||||||
|
k <- dr.k$k
|
||||||
|
V <- dr.k$V
|
||||||
|
q <- ncol(V)
|
||||||
|
# estimate bandwidth according alternative formula (see: TODO: see)
|
||||||
|
h <- estimate.bandwidth(X, k, sqrt(n), version = 2L)
|
||||||
|
# Projected `X`
|
||||||
|
XV <- X %*% V
|
||||||
|
# Devectorized distance matrix
|
||||||
|
# (inefficient in R but fast in C)
|
||||||
|
D <- matrix((XV[i, , drop = F] - XV[j, , drop = F])^2 %*% rep(1, q), n)
|
||||||
|
D <- D.eucl - D
|
||||||
|
# Apply kernel
|
||||||
|
K <- exp((-0.5 / h^2) * D^2)
|
||||||
|
# sum columns
|
||||||
|
colSumsK <- colSums(K)
|
||||||
|
# compute weighted and square meighted reponses
|
||||||
|
y1 <- (K %*% Y) / colSumsK
|
||||||
|
y2 <- (K %*% Y^2) / colSumsK
|
||||||
|
# element-wise L for dim. k
|
||||||
|
L[, as.character(k)] <- y2 - y1^2
|
||||||
|
}
|
||||||
|
|
||||||
|
for (ind in seq_len(length(object$res) - 1L)) {
|
||||||
|
p.test <- wilcox.test(L[, ind], L[, ind + 1L],
|
||||||
|
alternative = "less")$p.value
|
||||||
|
if (p.test < p.value) {
|
||||||
|
return(list(
|
||||||
|
p.value = p.test,
|
||||||
|
k = object$res[[ind]]$k
|
||||||
|
))
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
return(list(
|
||||||
|
p.value = NA,
|
||||||
|
k = object$res[[length(object$res)]]$k
|
||||||
|
))
|
||||||
|
}
|
||||||
|
|
||||||
|
#' Predicts SDR dimension using \code{\link[mda]{mars}} via a Cross-Validation.
|
||||||
|
#' TODO: rewrite!!!
|
||||||
|
#'
|
||||||
|
#' @param object instance of class \code{cve} (result of \code{cve},
|
||||||
|
#' \code{cve.call}).
|
||||||
|
#' @param ... ignored.
|
||||||
|
#'
|
||||||
|
#' @return list with
|
||||||
|
#' \itemize{
|
||||||
|
#' \item MSE: Mean Square Error,
|
||||||
|
#' \item k: predicted dimensions.
|
||||||
|
#' }
|
||||||
|
#'
|
||||||
|
#' @section cv:
|
||||||
|
#' Cross-validation ... TODO:
|
||||||
|
#'
|
||||||
|
#' @section elbow:
|
||||||
|
#' Cross-validation ... TODO:
|
||||||
|
#'
|
||||||
|
#' @section wilcoxon:
|
||||||
|
#' Cross-validation ... TODO:
|
||||||
|
#'
|
||||||
|
#' @examples
|
||||||
|
#' # create B for simulation
|
||||||
|
#' B <- rep(1, 5) / sqrt(5)
|
||||||
|
#'
|
||||||
|
#' set.seed(21)
|
||||||
|
#' # creat predictor data x ~ N(0, I_p)
|
||||||
|
#' x <- matrix(rnorm(500), 100)
|
||||||
|
#'
|
||||||
|
#' # simulate response variable
|
||||||
|
#' # y = f(B'x) + err
|
||||||
|
#' # with f(x1) = x1 and err ~ N(0, 0.25^2)
|
||||||
|
#' y <- x %*% B + 0.25 * rnorm(100)
|
||||||
|
#'
|
||||||
|
#' # Calculate cve for unknown k between min.dim and max.dim.
|
||||||
|
#' cve.obj.simple <- cve(y ~ x)
|
||||||
|
#'
|
||||||
|
#' predict_dim(cve.obj.simple)
|
||||||
|
#'
|
||||||
|
#' @export
|
||||||
|
predict_dim <- function(object, ..., method = "CV") {
|
||||||
|
# Check if there are dimensions to select.
|
||||||
|
if (length(object$res) == 1L) {
|
||||||
|
return(list(
|
||||||
|
message = "Only one dim. estimated.",
|
||||||
|
k = as.integer(names(object$res))
|
||||||
|
))
|
||||||
|
}
|
||||||
|
|
||||||
|
# Determine method "fuzzy".
|
||||||
|
methods <- c("cv", "elbow", "wilcoxon")
|
||||||
|
names(methods) <- methods
|
||||||
|
method <- methods[[tolower(method), exact = FALSE]]
|
||||||
|
if (is.null(method)) {
|
||||||
|
stop('Unable to determine method.')
|
||||||
|
}
|
||||||
|
|
||||||
|
if (method == "cv") {
|
||||||
|
return(predict_dim_cv(object))
|
||||||
|
} else if (method == "elbow") {
|
||||||
|
return(predict_dim_elbow(object))
|
||||||
|
} else if (method == "wilcoxon") {
|
||||||
|
return(predict_dim_wilcoxon(object))
|
||||||
|
} else {
|
||||||
|
stop("Unable to determine method.")
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
|
||||||
|
|
@ -0,0 +1,43 @@
|
||||||
|
% Generated by roxygen2: do not edit by hand
|
||||||
|
% Please edit documentation in R/predict_dim.R
|
||||||
|
\name{predict_dim}
|
||||||
|
\alias{predict_dim}
|
||||||
|
\title{Predicts SDR dimension using \code{\link[mda]{mars}} via a Cross-Validation.}
|
||||||
|
\usage{
|
||||||
|
predict_dim(object, ...)
|
||||||
|
}
|
||||||
|
\arguments{
|
||||||
|
\item{object}{instance of class \code{cve} (result of \code{cve},
|
||||||
|
\code{cve.call}).}
|
||||||
|
|
||||||
|
\item{...}{ignored.}
|
||||||
|
}
|
||||||
|
\value{
|
||||||
|
list with
|
||||||
|
\itemize{
|
||||||
|
\item MSE: Mean Square Error,
|
||||||
|
\item k: predicted dimensions.
|
||||||
|
}
|
||||||
|
}
|
||||||
|
\description{
|
||||||
|
Predicts SDR dimension using \code{\link[mda]{mars}} via a Cross-Validation.
|
||||||
|
}
|
||||||
|
\examples{
|
||||||
|
# create B for simulation
|
||||||
|
B <- rep(1, 5) / sqrt(5)
|
||||||
|
|
||||||
|
set.seed(21)
|
||||||
|
# creat predictor data x ~ N(0, I_p)
|
||||||
|
x <- matrix(rnorm(500), 100)
|
||||||
|
|
||||||
|
# simulate response variable
|
||||||
|
# y = f(B'x) + err
|
||||||
|
# with f(x1) = x1 and err ~ N(0, 0.25^2)
|
||||||
|
y <- x \%*\% B + 0.25 * rnorm(100)
|
||||||
|
|
||||||
|
# Calculate cve for unknown k between min.dim and max.dim.
|
||||||
|
cve.obj.simple <- cve(y ~ x)
|
||||||
|
|
||||||
|
predict_dim(cve.obj.simple)
|
||||||
|
|
||||||
|
}
|
||||||
|
|
@ -17,6 +17,7 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
double loss, loss_last, loss_best, err, tau;
|
double loss, loss_last, loss_best, err, tau;
|
||||||
double tol = tol_init * sqrt((double)(2 * q));
|
double tol = tol_init * sqrt((double)(2 * q));
|
||||||
double agility = -2.0 * (1.0 - momentum) / (h * h);
|
double agility = -2.0 * (1.0 - momentum) / (h * h);
|
||||||
|
double sumK;
|
||||||
double c = agility / (double)n;
|
double c = agility / (double)n;
|
||||||
|
|
||||||
// TODO: check parameters! dim, ...
|
// TODO: check parameters! dim, ...
|
||||||
|
|
@ -87,8 +88,9 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
S = laplace(adjacence(L, Y, y1, D, W, gauss, S), workMem);
|
S = laplace(adjacence(L, Y, y1, D, W, gauss, S), workMem);
|
||||||
} else if (method == weighted) {
|
} else if (method == weighted) {
|
||||||
colSumsK = elemApply(colSumsK, '-', 1.0, colSumsK);
|
colSumsK = elemApply(colSumsK, '-', 1.0, colSumsK);
|
||||||
loss_last = dot(L, '/', colSumsK);
|
sumK = sum(colSumsK);
|
||||||
c = agility / sum(colSumsK);
|
loss_last = dot(L, '*', colSumsK) / sumK;
|
||||||
|
c = agility / sumK;
|
||||||
/* Calculate the scaling matrix S */
|
/* Calculate the scaling matrix S */
|
||||||
S = laplace(adjacence(L, Y, y1, D, K, gauss, S), workMem);
|
S = laplace(adjacence(L, Y, y1, D, K, gauss, S), workMem);
|
||||||
} else {
|
} else {
|
||||||
|
|
@ -100,10 +102,8 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
G = matrixprod(c, tmp2, V, 0.0, G);
|
G = matrixprod(c, tmp2, V, 0.0, G);
|
||||||
|
|
||||||
if (logger) {
|
if (logger) {
|
||||||
callLogger(logger, loggerEnv,
|
callLogger(logger, loggerEnv, attempt, /* iter <- 0L */ -1,
|
||||||
attempt, /* iter <- 0L */ -1,
|
L, V, G, loss_last, /* err <- NA */ -1.0, tau);
|
||||||
L, V, G,
|
|
||||||
loss_last, /* err <- NA */ -1.0, tau);
|
|
||||||
}
|
}
|
||||||
|
|
||||||
/* Compute Skew-Symmetric matrix `A` used in Cayley transform.
|
/* Compute Skew-Symmetric matrix `A` used in Cayley transform.
|
||||||
|
|
@ -120,9 +120,6 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
/* Move `V` along the gradient direction. */
|
/* Move `V` along the gradient direction. */
|
||||||
V_tau = cayleyTransform(A, V, V_tau, workMem);
|
V_tau = cayleyTransform(A, V, V_tau, workMem);
|
||||||
|
|
||||||
// Rprintf("Start attempt(%2d), iter (%2d): err: %f, loss: %f, tau: %f\n",
|
|
||||||
// attempt, iter, dist(V, V_tau), loss_last, tau);
|
|
||||||
|
|
||||||
/* Embed X_i's in V space */
|
/* Embed X_i's in V space */
|
||||||
XV = matrixprod(1.0, X, V_tau, 0.0, XV);
|
XV = matrixprod(1.0, X, V_tau, 0.0, XV);
|
||||||
/* Compute embedded distances */
|
/* Compute embedded distances */
|
||||||
|
|
@ -146,7 +143,8 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
loss = mean(L);
|
loss = mean(L);
|
||||||
} else if (method == weighted) {
|
} else if (method == weighted) {
|
||||||
colSumsK = elemApply(colSumsK, '-', 1.0, colSumsK);
|
colSumsK = elemApply(colSumsK, '-', 1.0, colSumsK);
|
||||||
loss = dot(L, '/', colSumsK);
|
sumK = sum(colSumsK);
|
||||||
|
loss = dot(L, '*', colSumsK) / sumK;
|
||||||
} else {
|
} else {
|
||||||
// TODO: error handling!
|
// TODO: error handling!
|
||||||
}
|
}
|
||||||
|
|
@ -154,22 +152,26 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
/* Check if step is appropriate, iff not reduce learning rate. */
|
/* Check if step is appropriate, iff not reduce learning rate. */
|
||||||
if ((loss - loss_last) > loss_last * slack) {
|
if ((loss - loss_last) > loss_last * slack) {
|
||||||
tau *= gamma;
|
tau *= gamma;
|
||||||
|
iter -= 1;
|
||||||
A = elemApply(A, '*', gamma, A); // scale A by gamma
|
A = elemApply(A, '*', gamma, A); // scale A by gamma
|
||||||
continue;
|
continue;
|
||||||
|
} else {
|
||||||
|
tau /= gamma;
|
||||||
}
|
}
|
||||||
|
|
||||||
/* Compute error, use workMem. */
|
/* Compute error, use workMem. */
|
||||||
err = dist(V, V_tau);
|
err = dist(V, V_tau);
|
||||||
|
|
||||||
|
// Rprintf("%2d - iter: %2d, loss: %1.3f, err: %1.3f, tau: %1.3f, norm(G) = %1.3f\n",
|
||||||
|
// attempt, iter, loss, err, tau, sqrt(squareSum(G)));
|
||||||
|
|
||||||
/* Shift next step to current step and store loss to last. */
|
/* Shift next step to current step and store loss to last. */
|
||||||
V = copy(V_tau, V);
|
V = copy(V_tau, V);
|
||||||
loss_last = loss;
|
loss_last = loss;
|
||||||
|
|
||||||
if (logger) {
|
if (logger) {
|
||||||
callLogger(logger, loggerEnv,
|
callLogger(logger, loggerEnv, attempt, iter,
|
||||||
attempt, iter,
|
L, V, G, loss, err, tau);
|
||||||
L, V, G,
|
|
||||||
loss, err, tau);
|
|
||||||
}
|
}
|
||||||
|
|
||||||
/* Check Break condition. */
|
/* Check Break condition. */
|
||||||
|
|
@ -183,7 +185,7 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
} else if (method == weighted) {
|
} else if (method == weighted) {
|
||||||
/* Calculate the scaling matrix S */
|
/* Calculate the scaling matrix S */
|
||||||
S = laplace(adjacence(L, Y, y1, D, K, gauss, S), workMem);
|
S = laplace(adjacence(L, Y, y1, D, K, gauss, S), workMem);
|
||||||
c = agility / sum(colSumsK);
|
c = agility / sumK; // n removed previousely
|
||||||
} else {
|
} else {
|
||||||
// TODO: error handling!
|
// TODO: error handling!
|
||||||
}
|
}
|
||||||
|
|
@ -198,6 +200,8 @@ void cve(const mat *X, const mat *Y, const double h,
|
||||||
A = skew(tau, G, V, 0.0, A);
|
A = skew(tau, G, V, 0.0, A);
|
||||||
}
|
}
|
||||||
|
|
||||||
|
// Rprintf("\n");
|
||||||
|
|
||||||
/* Check if current attempt improved previous ones */
|
/* Check if current attempt improved previous ones */
|
||||||
if (attempt == 0 || loss < loss_best) {
|
if (attempt == 0 || loss < loss_best) {
|
||||||
loss_best = loss;
|
loss_best = loss;
|
||||||
|
|
|
||||||
381
LaTeX/notes.tex
381
LaTeX/notes.tex
|
|
@ -4,58 +4,369 @@
|
||||||
\usepackage[T1]{fontenc}
|
\usepackage[T1]{fontenc}
|
||||||
\usepackage{amsmath, amsfonts, amssymb, amsthm}
|
\usepackage{amsmath, amsfonts, amssymb, amsthm}
|
||||||
\usepackage{tikz}
|
\usepackage{tikz}
|
||||||
|
\usepackage{listings}
|
||||||
\usepackage{fullpage}
|
\usepackage{fullpage}
|
||||||
|
|
||||||
|
|
||||||
|
\lstdefinelanguage{PseudoCode} {
|
||||||
|
morekeywords={
|
||||||
|
for,
|
||||||
|
while,
|
||||||
|
repeat,
|
||||||
|
from,
|
||||||
|
each,
|
||||||
|
foreach,
|
||||||
|
break,
|
||||||
|
continue,
|
||||||
|
in,
|
||||||
|
do,
|
||||||
|
as,
|
||||||
|
and,
|
||||||
|
or,
|
||||||
|
end,
|
||||||
|
return,
|
||||||
|
if,
|
||||||
|
then,
|
||||||
|
else,
|
||||||
|
function,
|
||||||
|
begin,
|
||||||
|
to,
|
||||||
|
new,
|
||||||
|
input,
|
||||||
|
output
|
||||||
|
},
|
||||||
|
morecomment=[l]{/*},
|
||||||
|
morecomment=[l]{//},
|
||||||
|
% basicstyle=\ttfamily,
|
||||||
|
% keywordstyle=\color{blue}, %\ttfamily,
|
||||||
|
commentstyle=\color{gray}\it,
|
||||||
|
keywordstyle=\bf,
|
||||||
|
rulecolor=\color{black},
|
||||||
|
literate=%
|
||||||
|
{!=}{{$\neq$}}1
|
||||||
|
{<=}{{$\leq$}}1
|
||||||
|
{>=}{{$\geq$}}1
|
||||||
|
{->}{{$\rightarrow$}}1
|
||||||
|
{<-}{{$\leftarrow$}}1
|
||||||
|
}
|
||||||
|
|
||||||
|
% },
|
||||||
|
% tabsize=3,
|
||||||
|
% sensitive=false,
|
||||||
|
% morecomment=[l]{#},
|
||||||
|
% morestring=[b]",
|
||||||
|
% extendedchars=true,
|
||||||
|
% inputencoding=utf8,
|
||||||
|
% literate=%
|
||||||
|
% {!=}{{$\neq$}}1
|
||||||
|
% {<=}{{$\leq$}}1
|
||||||
|
% {>=}{{$\geq$}}1
|
||||||
|
% {<>}{{$\neq$}}1
|
||||||
|
% {:=}{{$\ \leftarrow\quad$}}1
|
||||||
|
% {Ö}{{\"O}}1
|
||||||
|
% {Ä}{{\"A}}1
|
||||||
|
% {Ü}{{\"U}}1
|
||||||
|
% {ß}{{\ss{}}}1
|
||||||
|
% {ü}{{\"u}}1
|
||||||
|
% {ä}{{\"a}}1
|
||||||
|
% {ö}{{\"o}}1
|
||||||
|
% {~}{{\textasciitilde}}1,
|
||||||
|
% texcl=true % use all chars from \usepackage[utf8]{inputenc}
|
||||||
|
% }
|
||||||
|
\lstset{
|
||||||
|
tabsize=4,
|
||||||
|
xleftmargin=0pt, % left margin
|
||||||
|
numbers=left, % linenumber position
|
||||||
|
numbersep=15pt, % left linenumber padding
|
||||||
|
numberstyle=\tiny,
|
||||||
|
basicstyle=\ttfamily,
|
||||||
|
keywordstyle=\color{black!60},
|
||||||
|
commentstyle=\ttfamily\color{gray!70},
|
||||||
|
breaklines=true,
|
||||||
|
literate=
|
||||||
|
}
|
||||||
|
|
||||||
|
\renewcommand{\epsilon}{\varepsilon}
|
||||||
|
|
||||||
\newcommand{\vecl}{\ensuremath{\operatorname{vec}_l}}
|
\newcommand{\vecl}{\ensuremath{\operatorname{vec}_l}}
|
||||||
\newcommand{\Sym}{\ensuremath{\operatorname{Sym}}}
|
\newcommand{\Sym}{\ensuremath{\operatorname{Sym}}}
|
||||||
|
|
||||||
|
\renewcommand{\vec}{\operatorname{vec}}
|
||||||
|
\newcommand{\devec}{\operatorname*{devec}}
|
||||||
|
\newcommand{\svec}{\operatorname{svec}}
|
||||||
|
\newcommand{\sym}{\operatorname{sym}}
|
||||||
|
\renewcommand{\skew}{\operatorname{skew}}
|
||||||
|
\newcommand{\rowSums}{\operatorname{rowSums}}
|
||||||
|
\newcommand{\colSums}{\operatorname{colSums}}
|
||||||
|
\newcommand{\diag}{\operatorname{diag}}
|
||||||
|
|
||||||
\begin{document}
|
\begin{document}
|
||||||
|
|
||||||
Indexing a given matrix $A = (a_{ij})_{i,j = 1, ..., n} \in \mathbb{R}^{n\times n}$ given as
|
\section{Kronecker Product Properties}
|
||||||
|
The \emph{mixed-product} property for matrices $A, B, C, D$ holds if and only if the following matrix products are well defined
|
||||||
\begin{displaymath}
|
\begin{displaymath}
|
||||||
A = \begin{pmatrix}
|
(A\otimes B)(C \otimes D) = (A C) \otimes (B C).
|
||||||
a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,n-1} \\
|
\end{displaymath}
|
||||||
a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,n-1} \\
|
In combination with the \emph{Hadamard product} (element-wise multiplication) for matrices $A, C$ of the same size as well as $B, D$ of the same size is
|
||||||
a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,n-1} \\
|
\begin{displaymath}
|
||||||
\vdots & \vdots & \vdots & \ddots & \vdots \\
|
(A\otimes B)\circ (C \otimes D) = (A \circ C) \otimes (B \circ D).
|
||||||
a_{n-1,0} & a_{n-1,1} & a_{n-1,2} & \ldots & a_{n-1,n-1}
|
\end{displaymath}
|
||||||
\end{pmatrix}
|
The \emph{transpose} of the Kronecker product fulfills
|
||||||
|
\begin{displaymath}
|
||||||
|
(A\otimes B)^T = A^T \otimes B^T
|
||||||
\end{displaymath}
|
\end{displaymath}
|
||||||
|
|
||||||
A symmetric matrix with zero main diagonal, meaning a matrix $S = S^T$ with $S_{i,i} = 0,\ \forall i = 1,..,n$ is givne in the following form
|
\section{Distance Computation}
|
||||||
\begin{displaymath}
|
The pair-wise distances $d_V(X_{i,:}, X_{j,:})$ arranged in the distance matrix $D\in\mathbb{R}^{n\times n}$ can be written as
|
||||||
S = \begin{pmatrix}
|
\begin{align*}
|
||||||
0 & s_{1,0} & s_{2,0} & \ldots & s_{n-1,0} \\
|
\vec(D) = \rowSums(((X Q)\otimes 1_n - 1_n \otimes (X Q))^2)
|
||||||
s_{1,0} & 0 & s_{2,1} & \ldots & s_{n-1,1} \\
|
\end{align*}
|
||||||
s_{2,0} & s_{2,1} & 0 & \ldots & s_{n-1,2} \\
|
This can be computed in $\mathcal{O}(n^2p + np^2)$ time (vectorization and devectorization takes $\mathcal{O}(1)$).
|
||||||
\vdots & \vdots & \vdots & \ddots & \vdots \\
|
|
||||||
s_{n-1,0} & s_{n-1,1} & s_{n-1,2} & \ldots & 0
|
|
||||||
\end{pmatrix}
|
|
||||||
\end{displaymath}
|
|
||||||
Therefore its sufficient to store only the lower triangular part, for memory efficiency and some further alrogithmic shortcuts (sometime they are more expencife) the symmetric matrix $S$ is stored in packed form, meanin in a vector of the length $\frac{n(n-1)}{2}$. We use (like for matrices) a column-major order of elements and define the $\vecl:\Sym(n)\to \mathbb{R}^{n(n-1) / 2}$ opperator defined as
|
|
||||||
|
|
||||||
|
The matrices $K, W$ are define through there elements as
|
||||||
\begin{displaymath}
|
\begin{displaymath}
|
||||||
\vecl(S) = (s_{1,0}, s_{2,0},\cdots,s_{n-1,0},s_{2,1}\cdots,s_{n-1,n-2})^T
|
k_{i j} = \exp\left(-\frac{d_{i j}^2}{2 h^2}\right),\qquad w_{i j} = \frac{k_{i j}}{\sum_{m} k_{m j}}.
|
||||||
\end{displaymath}
|
\end{displaymath}
|
||||||
|
|
||||||
The relation between the matrix indices $i,j$ and the $\vecl$ index $k$ is given by
|
Next are $\bar{y}^{(m)}$ and the ``element-wise'' loss $l_i = L_n(V, X_i)$.
|
||||||
|
|
||||||
\begin{displaymath}
|
\begin{displaymath}
|
||||||
(\vecl(S)_k = s_{i,j} \quad\Leftrightarrow\quad k = jn+i) : j \in \{0,...,n-2\} \land j < i < n.
|
\bar{y}^{(m)} = W^T Y^m,\qquad l = \bar{y}^{(2)} - (\bar{y}^{(1)})^2
|
||||||
\end{displaymath}
|
\end{displaymath}
|
||||||
|
|
||||||
\begin{center}
|
\section{Gradient Computation}
|
||||||
\begin{tikzpicture}[xscale=1,yscale=-1]
|
The model
|
||||||
% \foreach \i in {0,...,5} {
|
\begin{displaymath}
|
||||||
% \node at ({mod(\i, 3)}, {int(\i / 3)}) {$\i$};
|
Y \sim g(B^T X) + \epsilon.
|
||||||
% }
|
\end{displaymath}
|
||||||
\foreach \i in {1,...,4} {
|
|
||||||
\foreach \j in {1,...,\i} {
|
Assume a data set $(X_i, Y_i)$ for $i = 1, ..., n$ with $X$ a $n\times p$ matrix such that each row represents one sample. Now let $l_i = L_n(V, X_i)$, $\bar{y}^{(1)}_j = (W^T Y)_j$ as well as $d_{i j}, w_{i j}$ the distance and weight matrix components. Then the gradient for the ``simple'' CVE method is given as
|
||||||
\node at (\j, \i) {$\i,\j$};
|
\begin{displaymath}
|
||||||
}
|
\nabla L_n(V) = \frac{1}{nh^2}\sum_{i = 1}^{n} \sum_{j = 1}^{n} (l_j - (Y_i - \bar{y}^{(1)}_j)^2) w_{i j} d_{i j} \nabla_V d_V(X_{i,:}, X_{j,:}).
|
||||||
}
|
\end{displaymath}
|
||||||
|
This representation is cumbersome and a direct implementation has a asymptotic run-time of $\Theta(n^2p^2)$ because it is a double sum over $n$, therefore quadratic in $n$, and the form of $\nabla_V d_V$.
|
||||||
|
|
||||||
|
This can be optimized and written in matrix notation. First the distance gradient is given as
|
||||||
|
\begin{displaymath}
|
||||||
|
\nabla_V d_V(X_{i,:}, X_{j,:}) = -2 (X_{i,:} - X_{j,:})^T (X_{i,:} - X_{j,:}) V
|
||||||
|
\end{displaymath}
|
||||||
|
(Note: $X_{i,:}\in\mathbb{R}^{1\times p}$, aka a row representing one sample). In addition define the $n\times n$ matrix $S$ through its elements
|
||||||
|
\begin{displaymath}
|
||||||
|
s_{i j} = (l_j - (Y_i - \bar{y}^{(1)}_j)^2) w_{i j} d_{i j}.
|
||||||
|
\end{displaymath}
|
||||||
|
Substitution in the gradient leads to
|
||||||
|
\begin{align*}
|
||||||
|
\nabla L_n(V)
|
||||||
|
&= -\frac{2}{nh^2}\sum_{i = 1}^{n} \sum_{j = 1}^{n} s_{i j} (X_{i,:} - X_{j,:})^T (X_{i,:} - X_{j,:}) V \\
|
||||||
|
&= -\frac{2}{nh^2}\sum_{i = 1}^{n} \sum_{j = 1}^{n} s_{i j} \left( X_{i,:}^T X_{i,:} - X_{i,:}^T X_{j,:} - X_{j,:}^T X_{i,:} + X_{j,:}^T X_{j,:} \right) V \\
|
||||||
|
&= -\frac{2}{nh^2} \left( \sum_{i = 1}^{n}\sum_{j = 1}^{n} (s_{i j} + s_{j i}) X_{i,:}^T X_{i,:} - \sum_{i = 1}^{n}\sum_{j = 1}^{n} (s_{i j} + s_{j i}) X_{i,:}^T X_{j,:} \right) V \\
|
||||||
|
&= -\frac{2}{nh^2} \left( X^T \diag(\colSums(S + S^T)) X - X^T (S + S^T) X \right) V \\
|
||||||
|
&= -\frac{2}{nh^2} X^T \left( \diag(\colSums(S + S^T)) - (S + S^T) \right) X V
|
||||||
|
\end{align*}
|
||||||
|
|
||||||
|
\begin{center}{\bf
|
||||||
|
ATTENTION: The given R examples are to illustrate the inplementation in C which is 0-indexed!
|
||||||
|
}\end{center}
|
||||||
|
|
||||||
|
The \emph{vertorization} operation maps a matrix $A\in\mathbb{R}^{n\times m}$ into $\mathbb{R}^{nm}$ by stacking the columns of $A$;
|
||||||
|
\begin{displaymath}
|
||||||
|
\vec(A) = (a_{0,0}, a_{0,1}, a_{0,2},...,a_{0,n-1},a_{1,0},a_{1,1},...,a_{n-1,n-1})^T.
|
||||||
|
\end{displaymath}
|
||||||
|
The relation $\vec(A)_k = a_{i,j}$ holds for $k=nj+i$ such that $0\leq k < n^2$ and $0\leq i < n, 0 \leq j < m$. This operation is obviously a bijection. When going ``backwards'' the dimension of the original space is required, therefore let $\devec_n$ be the operation such that $\devec_n(\vec(A)) = A$ for $A\in\mathbb{R}^{n\times m}$.\footnote{Note that for $B\in\mathbb{R}^{p\times q}$ such that $pq = nm$ the $\devec_n(\vec(B))\in\mathbb{R}^{n\times m}$.}
|
||||||
|
|
||||||
|
For symmetric matrices the information stored in $a_{i,j} = a_{j,i}$ is twice stored in $A=A^T\in\mathbb{R}^{n\times n}$, to remove this redundency the \emph{symmetric vectorization} is defined which saves the main diagonal and the lower triangular part of the symmetric matrix according the scema
|
||||||
|
\begin{displaymath}
|
||||||
|
\svec(A) = (a_{0,0},2a_{1,0},2a_{2,n},...,2a_{n-1,0},a_{1,1},2a_{2,1},...,2a_{n-1,1},a_{2,2},...,a_{n-1,n-1})
|
||||||
|
\end{displaymath}
|
||||||
|
A it more formal
|
||||||
|
\begin{displaymath}
|
||||||
|
\svec(A)_{k} = (2-\delta_{i,j})a_{i,j} \quad\text{for}\quad k = n j + i - \frac{j(j + 1)}{2}, 0\leq j \leq i < n^2.
|
||||||
|
\end{displaymath}
|
||||||
|
|
||||||
|
\begin{lstlisting}[language=R]
|
||||||
|
n <- 3
|
||||||
|
k <- function(i, j, n) { (j * n) + i - (j * (j + 1) / 2) }
|
||||||
|
i <- function(n) { rep(1:n - 1, n) }
|
||||||
|
j <- function(n) { rep(1:n - 1, each = n) }
|
||||||
|
A <- matrix(k(i(n), j(n), n), n)
|
||||||
|
A[which(j(n) > i(n))] <- NA
|
||||||
|
A
|
||||||
|
# [,1] [,2] [,3]
|
||||||
|
# [1,] 0 NA NA
|
||||||
|
# [2,] 1 3 NA
|
||||||
|
# [3,] 2 4 5
|
||||||
|
vec <- function(A) { as.vector(A) }
|
||||||
|
svec <- function(A) {
|
||||||
|
n <- nrow(A)
|
||||||
|
((2 - (i(n) == j(n))) * A)[i(n) >= j(n)]
|
||||||
|
}
|
||||||
|
svec(matrix(1, n, n))
|
||||||
|
# [1] 1 2 2 1 2 1
|
||||||
|
devec <- function(vec, n) { matrix(vec, n) }
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
|
For a quadratic matrix $A\in\mathbb{R}^{n\times n}$ we define
|
||||||
|
\begin{displaymath}
|
||||||
|
\sym(A) := \frac{A + A^T}{2}, \qquad \skew(A) := \frac{A - A^T}{2}.
|
||||||
|
\end{displaymath}
|
||||||
|
|
||||||
|
% For a Matrix $A\in\mathbb{R}^{n\times n}$ the \emph{vectorization} operation is defined as a mapping from the matrices into a
|
||||||
|
|
||||||
|
% Indexing a given matrix $A = (a_{ij})_{i,j = 1, ..., n} \in \mathbb{R}^{n\times n}$ given as
|
||||||
|
% \begin{displaymath}
|
||||||
|
% A = \begin{pmatrix}
|
||||||
|
% a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,n-1} \\
|
||||||
|
% a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,n-1} \\
|
||||||
|
% a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,n-1} \\
|
||||||
|
% \vdots & \vdots & \vdots & \ddots & \vdots \\
|
||||||
|
% a_{n-1,0} & a_{n-1,1} & a_{n-1,2} & \ldots & a_{n-1,n-1}
|
||||||
|
% \end{pmatrix}
|
||||||
|
% \end{displaymath}
|
||||||
|
|
||||||
|
% A symmetric matrix with zero main diagonal, meaning a matrix $S = S^T$ with $S_{i,i} = 0,\ \forall i = 1,..,n$ is given in the following form
|
||||||
|
% \begin{displaymath}
|
||||||
|
% S = \begin{pmatrix}
|
||||||
|
% 0 & s_{1,0} & s_{2,0} & \ldots & s_{n-1,0} \\
|
||||||
|
% s_{1,0} & 0 & s_{2,1} & \ldots & s_{n-1,1} \\
|
||||||
|
% s_{2,0} & s_{2,1} & 0 & \ldots & s_{n-1,2} \\
|
||||||
|
% \vdots & \vdots & \vdots & \ddots & \vdots \\
|
||||||
|
% s_{n-1,0} & s_{n-1,1} & s_{n-1,2} & \ldots & 0
|
||||||
|
% \end{pmatrix}
|
||||||
|
% \end{displaymath}
|
||||||
|
% Therefore its sufficient to store only the lower triangular part, for memory efficiency and some further algorithmic shortcuts (sometime they are more expensive) the symmetric matrix $S$ is stored in packed form, meaning in a vector of the length $\frac{n(n-1)}{2}$. We use (like for matrices) a column-major order of elements and define the $\vecl:\Sym(n)\to \mathbb{R}^{n(n-1) / 2}$ operator defined as
|
||||||
|
|
||||||
|
% \begin{displaymath}
|
||||||
|
% \vecl(S) = (s_{1,0}, s_{2,0},\cdots,s_{n-1,0},s_{2,1}\cdots,s_{n-1,n-2})^T
|
||||||
|
% \end{displaymath}
|
||||||
|
|
||||||
|
% The relation between the matrix indices $i,j$ and the $\vecl$ index $k$ is given by
|
||||||
|
|
||||||
|
% \begin{displaymath}
|
||||||
|
% (\vecl(S)_k = s_{i,j} \quad\Leftrightarrow\quad k = jn+i) : j \in \{0,...,n-2\} \land j < i < n.
|
||||||
|
% \end{displaymath}
|
||||||
|
|
||||||
|
% \begin{center}
|
||||||
|
% \begin{tikzpicture}[xscale=1,yscale=-1]
|
||||||
|
% % \foreach \i in {0,...,5} {
|
||||||
|
% % \node at ({mod(\i, 3)}, {int(\i / 3)}) {$\i$};
|
||||||
|
% % }
|
||||||
|
% \foreach \i in {1,...,4} {
|
||||||
|
% \foreach \j in {1,...,\i} {
|
||||||
|
% \node at (\j, \i) {$\i,\j$};
|
||||||
|
% }
|
||||||
|
% }
|
||||||
|
|
||||||
|
% \end{tikzpicture}
|
||||||
|
% \end{center}
|
||||||
|
|
||||||
|
\newpage
|
||||||
|
\section{Algorithm}
|
||||||
|
The basic algorithm reads as follows:
|
||||||
|
|
||||||
|
Mit
|
||||||
|
\begin{displaymath}
|
||||||
|
X_{diff} := X\otimes 1_n - 1_n\otimes X
|
||||||
|
\end{displaymath}
|
||||||
|
gilt
|
||||||
|
\begin{displaymath}
|
||||||
|
X_{diff}Q := (X\otimes 1_n - 1_n\otimes X)Q = XQ\otimes 1_n - 1_n\otimes XQ
|
||||||
|
\end{displaymath}
|
||||||
|
|
||||||
|
\newcommand{\rStiefel}{\operatorname{rStiefel}}
|
||||||
|
% \lstset{language=PseudoCode}
|
||||||
|
% \begin{lstlisting}[mathescape, caption=Erste Phase von \texttt{HDE} (siehe \cite{HDE}), label=code:HDE, captionpos=b]
|
||||||
|
% \begin{lstlisting}[mathescape]
|
||||||
|
% // Hallo Welt
|
||||||
|
% /* Hallo comment */
|
||||||
|
% $X_{diff} \leftarrow X\otimes 1_n - 1_n\otimes X$
|
||||||
|
|
||||||
|
% for attempt from 1 to attempts do
|
||||||
|
% if $\exists V_{init}$ then
|
||||||
|
% $V \leftarrow V_{init}$
|
||||||
|
% else
|
||||||
|
% $V \leftarrow \rStiefel(p, q)$
|
||||||
|
% end if
|
||||||
|
|
||||||
|
% /* Projection matrix into null space */
|
||||||
|
% $Q \leftarrow I_p - VV^T$
|
||||||
|
|
||||||
|
% /* Pair-wise distances (row sum of squared elements) */
|
||||||
|
% $D \leftarrow$ foreach $i,j=1,...,n$ as $D_{i,j}\leftarrow \|(X_{i,:}-X_{j,:})Q\|_2^2$
|
||||||
|
|
||||||
|
% /* Weights */
|
||||||
|
% $W \leftarrow$ foreach $i,j=1,...,n$ as $W_{i,j} \leftarrow \frac{k(D_{i,j})}{\sum_{i} k(D_{i,j})}$
|
||||||
|
|
||||||
|
% $\bar{y}_1 \leftarrow W^TY$
|
||||||
|
% $\bar{y}_2 \leftarrow W^T(Y\odot Y)$
|
||||||
|
|
||||||
|
% /* Element-wise losses */
|
||||||
|
% $L \leftarrow \bar{y}_2 - \bar{y}_1^2$
|
||||||
|
|
||||||
|
% for epoch from 1 to epochs do
|
||||||
|
|
||||||
|
% $G_t \leftarrow \gamma G_{t-1} + (1-\gamma) \nabla_c L(V)$
|
||||||
|
|
||||||
|
% end for
|
||||||
|
% end for
|
||||||
|
% \end{lstlisting}
|
||||||
|
|
||||||
|
The loss at a given position is
|
||||||
|
\begin{displaymath}
|
||||||
|
L_n(V) = \frac{1}{nh^2}\sum_{i = 0}^{n - 1} \sum_{j = 0}^{n - 1} (L_j - (Y_i - \bar{y}^{(1)}_j)^2) w_{i j} d_{i j} \nabla_V d_V(X_{i,:}, X_{j,:})
|
||||||
|
\end{displaymath}
|
||||||
|
Now let the matrix $S$ be defined through its coefficients
|
||||||
|
\begin{displaymath}
|
||||||
|
s_{i j} = (L_j - (Y_i - \bar{y}^{(1)}_j)^2) w_{i j} d_{i j}
|
||||||
|
\end{displaymath}
|
||||||
|
This matrix is \underline{not} symmetric but we can consider the symmetric $S + S^T$ with a zero main diagonal because $D$ has a zero main diagonal, meaning $s_{i i} = 0$ because $d_{i i} = 0$ for each $i$. Therefore the following holds due to the fact that $\nabla_V d_V(X_{i,:}, X_{j,:}) = \nabla_V d_V(X_{j,:}, X_{i,:})$.
|
||||||
|
\begin{displaymath}
|
||||||
|
L_n(V) = \frac{1}{nh^2}\sum_{j = 0}^{n - 1} \sum_{i = j}^{n - 1} (s_{i j} + s_{j i}) \nabla_V d_V(X_{i,:}, X_{j,:})
|
||||||
|
\end{displaymath}
|
||||||
|
Note the summation indices $0 \leq j \leq i < n$. Substitution with $\nabla_V d_V(X_{i,:}, X_{j,:}) = -2 (X_{i,:} - X_{j,:})^T(X_{i,:} - X_{j,:}) V$ evaluates to
|
||||||
|
\begin{displaymath}
|
||||||
|
L_n(V) = -\frac{2}{nh^2}\sum_{j = 0}^{n - 1} \sum_{i = j}^{n - 1} (s_{i j} + s_{j i}) (X_{i,:} - X_{j,:})^T(X_{i,:} - X_{j,:}) V
|
||||||
|
\end{displaymath}
|
||||||
|
Let $X_{-}$ be the matrix containing all pairs of $X_{i,:}$ to $X_{j,:}$ differences using the same row indexing scheme as the symmetric vectorization.
|
||||||
|
\begin{displaymath}
|
||||||
|
(X_{-})_{k,:} = X_{i,:} - X_{j,:} \quad\text{for}\quad k = n j + i - \frac{j(j + 1)}{2}, 0\leq j \leq i < n^2
|
||||||
|
\end{displaymath}
|
||||||
|
With the $X_{-}$ matrix the above double sum can be formalized in matrix notation as follows\footnote{only valid cause $s_{i i} = 0$}
|
||||||
|
\begin{displaymath}
|
||||||
|
L_n(V) = -\frac{2}{nh^2} X_{-}^T(\svec(\sym(S)) \circ_r X_{-}) V
|
||||||
|
\end{displaymath}
|
||||||
|
where $\circ_r$ means the ``recycled'' hadamard product, this is for a vector $x\in\mathbb{R}^n$ and a Matrix $M\in\mathbb{R}^{n\times m}$ just the element wise multiplication for each column of $M$ with $x$, or equivalent $x\circ_r M = \underbrace{(x, x, ..., x)}_{{n\times m}} \circ M$ where $\circ$ is the element-wise product.
|
||||||
|
|
||||||
|
|
||||||
|
\begin{lstlisting}[mathescape, language=PseudoCode]
|
||||||
|
/* Starting value and initial gradient. */
|
||||||
|
$V_1 \leftarrow V_{init}$ if $\exists V_{init}$ else $\rStiefel(p, q)$
|
||||||
|
$G_1 \leftarrow (1 - \mu) \nabla L_n(V_0)$
|
||||||
|
|
||||||
|
/* Optimization loop */
|
||||||
|
$t \leftarrow 1$
|
||||||
|
while $t\leq\,$max.iter do
|
||||||
|
|
||||||
|
/* Update on stiefel manifold. */
|
||||||
|
$A \leftarrow G_tV_t^T - V_tG_t^T$
|
||||||
|
$V_{t+1} \leftarrow (I_p + \tau A)^{-1}(I_p - \tau A)V_{t}$
|
||||||
|
|
||||||
|
/* Check break condition. */
|
||||||
|
if $\|V_{t+1}V_{t+1}^T - V_{t}^TV_{t}\|_2^2 \leq \sqrt{2q}\,$tol then
|
||||||
|
break
|
||||||
|
end if
|
||||||
|
|
||||||
|
/* Check for decrease. */
|
||||||
|
if $L_n(V_{t+1}) - L_n(V_{t}) > L_n(V_{t})\,$slack then // TODO: slack?
|
||||||
|
/* Reduce step-size. */
|
||||||
|
$\tau \leftarrow \gamma\tau$
|
||||||
|
else
|
||||||
|
/* Gradient at next position (with momentum). */
|
||||||
|
$G_{t+1} \leftarrow \mu G_{t} + (1 - \mu) \nabla L_n(V_{t+1})$
|
||||||
|
/* Increase step index */
|
||||||
|
$t \leftarrow t + 1$
|
||||||
|
end if
|
||||||
|
|
||||||
|
end while
|
||||||
|
\end{lstlisting}
|
||||||
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{center}
|
|
||||||
|
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|
@ -1,13 +1,31 @@
|
||||||
# Usage:
|
# Usage:
|
||||||
# ~$ Rscript runtime_test.R
|
# ~$ Rscript runtime_test.R
|
||||||
|
|
||||||
|
textplot <- function(...) {
|
||||||
|
text <- unlist(list(...))
|
||||||
|
if (length(text) > 20) {
|
||||||
|
text <- c(text[1:17],
|
||||||
|
' ...... (skipped, text too long) ......',
|
||||||
|
text[c(-1, 0) + length(text)])
|
||||||
|
}
|
||||||
|
|
||||||
|
plot(NA, xlim = c(0, 1), ylim = c(0, 1),
|
||||||
|
bty = 'n', xaxt = 'n', yaxt = 'n', xlab = '', ylab = '')
|
||||||
|
|
||||||
|
for (i in seq_along(text)) {
|
||||||
|
text(0, 1 - (i / 20),
|
||||||
|
text[[i]], pos = 4)
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
# library(CVEpureR) # load CVE's pure R implementation
|
# library(CVEpureR) # load CVE's pure R implementation
|
||||||
library(CVE) # load CVE
|
library(CVE) # load CVE
|
||||||
|
|
||||||
#' Writes log information to console. (to not get bored^^)
|
#' Writes log information to console. (to not get bored^^)
|
||||||
tell.user <- function(name, start.time, i, length) {
|
tell.user <- function(name, start, i, length) {
|
||||||
cat("\rRunning Test (", name, "):",
|
cat("\rRunning Test (", name, "):",
|
||||||
i, "/", length,
|
i, "/", length,
|
||||||
" - elapsed:", format(Sys.time() - start.time), "\033[K")
|
" - elapsed:", format(Sys.time() - start), "\033[K")
|
||||||
}
|
}
|
||||||
#' Computes "distance" of spanned subspaces.
|
#' Computes "distance" of spanned subspaces.
|
||||||
#' @param B1 Semi-orthonormal basis matrix
|
#' @param B1 Semi-orthonormal basis matrix
|
||||||
|
|
@ -29,19 +47,14 @@ MAXIT <- 50L
|
||||||
# number of arbitrary starting values for curvilinear optimization
|
# number of arbitrary starting values for curvilinear optimization
|
||||||
ATTEMPTS <- 10L
|
ATTEMPTS <- 10L
|
||||||
# set names of datasets
|
# set names of datasets
|
||||||
dataset.names <- c("M1", "M2", "M3", "M4", "M5")
|
ds.names <- paste0("M", seq(7))
|
||||||
# Set used CVE method
|
# Set used CVE method
|
||||||
methods <- c("simple") # c("legacy", "simple", "linesearch", "sgd")
|
methods <- c("simple", "weighted") # c("legacy", "simple", "linesearch", "sgd")
|
||||||
|
|
||||||
if ("legacy" %in% methods) {
|
|
||||||
# Source legacy code (but only if needed)
|
|
||||||
source("CVE_legacy/function_script.R")
|
|
||||||
}
|
|
||||||
|
|
||||||
# Setup error and time tracking variables
|
# Setup error and time tracking variables
|
||||||
error <- matrix(NA, SIM.NR, length(methods) * length(dataset.names))
|
error <- matrix(NA, SIM.NR, length(methods) * length(ds.names))
|
||||||
time <- matrix(NA, SIM.NR, ncol(error))
|
time <- matrix(NA, SIM.NR, ncol(error))
|
||||||
colnames(error) <- kronecker(paste0(dataset.names, '-'), methods, paste0)
|
colnames(error) <- kronecker(paste0(ds.names, '-'), methods, paste0)
|
||||||
colnames(time) <- colnames(error)
|
colnames(time) <- colnames(error)
|
||||||
|
|
||||||
# Create new log file and write CSV (actualy TSV) header.
|
# Create new log file and write CSV (actualy TSV) header.
|
||||||
|
|
@ -56,13 +69,12 @@ cat('Plotting to file:', path, '\n')
|
||||||
|
|
||||||
# only for telling user (to stdout)
|
# only for telling user (to stdout)
|
||||||
count <- 0
|
count <- 0
|
||||||
start.time <- Sys.time()
|
start <- Sys.time()
|
||||||
# Start simulation loop.
|
# Start simulation loop.
|
||||||
for (sim in 1:SIM.NR) {
|
for (sim in 1:SIM.NR) {
|
||||||
# Repeat for each dataset.
|
# Repeat for each dataset.
|
||||||
for (name in dataset.names) {
|
for (name in ds.names) {
|
||||||
count <- count + 1
|
tell.user(name, start, (count <- count + 1), SIM.NR * length(ds.names))
|
||||||
tell.user(name, start.time, count, SIM.NR * length(dataset.names))
|
|
||||||
|
|
||||||
# Create a new dataset
|
# Create a new dataset
|
||||||
ds <- dataset(name)
|
ds <- dataset(name)
|
||||||
|
|
@ -71,35 +83,20 @@ for (sim in 1:SIM.NR) {
|
||||||
X <- ds$X
|
X <- ds$X
|
||||||
data <- cbind(Y, X)
|
data <- cbind(Y, X)
|
||||||
# get dimensions
|
# get dimensions
|
||||||
dim <- ncol(X)
|
k <- ncol(ds$B)
|
||||||
truedim <- ncol(ds$B)
|
|
||||||
|
|
||||||
for (method in methods) {
|
for (method in methods) {
|
||||||
if (tolower(method) == "legacy") {
|
dr.time <- system.time(
|
||||||
dr.time <- system.time(
|
dr <- cve.call(X, Y,
|
||||||
dr <- stiefel_opt(data,
|
method = method,
|
||||||
k = dim - truedim,
|
k = k,
|
||||||
k0 = ATTEMPTS,
|
attempts = ATTEMPTS
|
||||||
h = estimate.bandwidth(X,
|
|
||||||
k = truedim,
|
|
||||||
nObs = sqrt(nrow(X))),
|
|
||||||
maxit = MAXIT
|
|
||||||
)
|
|
||||||
)
|
)
|
||||||
dr$B <- fill_base(dr$est_base)[, 1:truedim]
|
)
|
||||||
} else {
|
dr$B <- coef(dr, k)
|
||||||
dr.time <- system.time(
|
|
||||||
dr <- cve.call(X, Y,
|
|
||||||
method = method,
|
|
||||||
k = truedim,
|
|
||||||
attempts = ATTEMPTS
|
|
||||||
)
|
|
||||||
)
|
|
||||||
dr$B <- coef(dr, truedim)
|
|
||||||
}
|
|
||||||
|
|
||||||
key <- paste0(name, '-', method)
|
key <- paste0(name, '-', method)
|
||||||
error[sim, key] <- subspace.dist(dr$B, ds$B) / sqrt(2 * truedim)
|
error[sim, key] <- subspace.dist(dr$B, ds$B) / sqrt(2 * k)
|
||||||
time[sim, key] <- dr.time["elapsed"]
|
time[sim, key] <- dr.time["elapsed"]
|
||||||
|
|
||||||
# Log results to file (mostly for long running simulations)
|
# Log results to file (mostly for long running simulations)
|
||||||
|
|
|
||||||
63
test.R
63
test.R
|
|
@ -1,3 +1,19 @@
|
||||||
|
textplot <- function(...) {
|
||||||
|
text <- unlist(list(...))
|
||||||
|
if (length(text) > 20) {
|
||||||
|
text <- c(text[1:17],
|
||||||
|
' ...... (skipped, text too long) ......',
|
||||||
|
text[c(-1, 0) + length(text)])
|
||||||
|
}
|
||||||
|
|
||||||
|
plot(NA, xlim = c(0, 1), ylim = c(0, 1),
|
||||||
|
bty = 'n', xaxt = 'n', yaxt = 'n', xlab = '', ylab = '')
|
||||||
|
|
||||||
|
for (i in seq_along(text)) {
|
||||||
|
text(0, 1 - (i / 20),
|
||||||
|
text[[i]], pos = 4)
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
args <- commandArgs(TRUE)
|
args <- commandArgs(TRUE)
|
||||||
if (length(args) > 0L) {
|
if (length(args) > 0L) {
|
||||||
|
|
@ -10,11 +26,12 @@ if (length(args) > 1L) {
|
||||||
} else {
|
} else {
|
||||||
momentum <- 0.0
|
momentum <- 0.0
|
||||||
}
|
}
|
||||||
|
seed <- 42
|
||||||
max.iter <- 50L
|
max.iter <- 50L
|
||||||
attempts <- 25L
|
attempts <- 25L
|
||||||
|
|
||||||
library(CVE)
|
library(CVE)
|
||||||
path <- paste0('~/Projects/CVE/tmp/logger_', method, '_', momentum, '.C.pdf')
|
path <- paste0('~/Projects/CVE/tmp/logger_', method, '.C.pdf')
|
||||||
|
|
||||||
# Define logger for `cve()` method.
|
# Define logger for `cve()` method.
|
||||||
logger <- function(attempt, iter, data) {
|
logger <- function(attempt, iter, data) {
|
||||||
|
|
@ -29,12 +46,14 @@ logger <- function(attempt, iter, data) {
|
||||||
true.error.history[iter + 1, attempt] <<- true.error
|
true.error.history[iter + 1, attempt] <<- true.error
|
||||||
}
|
}
|
||||||
|
|
||||||
pdf(path)
|
pdf(path, width = 8.27, height = 11.7) # width, height unit is inces -> A4
|
||||||
par(mfrow = c(2, 2))
|
layout(matrix(c(1, 1,
|
||||||
|
2, 3,
|
||||||
|
4, 5), nrow = 3, byrow = TRUE))
|
||||||
|
|
||||||
for (name in paste0("M", seq(5))) {
|
for (name in paste0("M", seq(7))) {
|
||||||
# Seed random number generator
|
# Seed random number generator
|
||||||
set.seed(42)
|
set.seed(seed)
|
||||||
|
|
||||||
# Create a dataset
|
# Create a dataset
|
||||||
ds <- dataset(name)
|
ds <- dataset(name)
|
||||||
|
|
@ -52,11 +71,37 @@ for (name in paste0("M", seq(5))) {
|
||||||
tau.history <- matrix(NA, max.iter + 1, attempts)
|
tau.history <- matrix(NA, max.iter + 1, attempts)
|
||||||
true.error.history <- matrix(NA, max.iter + 1, attempts)
|
true.error.history <- matrix(NA, max.iter + 1, attempts)
|
||||||
|
|
||||||
dr <- cve(Y ~ X, k = k, method = method,
|
time <- system.time(
|
||||||
momentum = momentum,
|
dr <- cve(Y ~ X, k = k, method = method,
|
||||||
max.iter = max.iter, attempts = attempts,
|
momentum = momentum,
|
||||||
logger = logger)
|
max.iter = max.iter, attempts = attempts,
|
||||||
|
logger = logger)
|
||||||
|
)["elapsed"]
|
||||||
|
|
||||||
|
# Extract finaly selected values:
|
||||||
|
B.est <- coef(dr, k)
|
||||||
|
true.error <- norm(tcrossprod(B.est) - tcrossprod(B), 'F') / sqrt(2 * k)
|
||||||
|
loss <- dr$res[[as.character(k)]]$loss
|
||||||
|
|
||||||
|
# Write metadata.
|
||||||
|
textplot(
|
||||||
|
paste0("Seed value: ", seed),
|
||||||
|
"",
|
||||||
|
paste0("Dataset Name: ", ds$name),
|
||||||
|
paste0("dim(X) = (", nrow(X), ", ", ncol(X), ")"),
|
||||||
|
paste0("dim(B) = (", nrow(B), ", ", ncol(B), ")"),
|
||||||
|
"",
|
||||||
|
paste0("CVE method: ", dr$method),
|
||||||
|
paste0("Max Iterations: ", max.iter),
|
||||||
|
paste0("Attempts: ", attempts),
|
||||||
|
paste0("Momentum: ", momentum),
|
||||||
|
"CVE call:",
|
||||||
|
paste0(" > ", format(dr$call)),
|
||||||
|
"",
|
||||||
|
paste0("True Error: ", round(true.error, 3)),
|
||||||
|
paste0("loss: ", round(loss, 3)),
|
||||||
|
paste0("time: ", round(time, 3), " s")
|
||||||
|
)
|
||||||
# Plot history's
|
# Plot history's
|
||||||
matplot(loss.history, type = 'l', log = 'y', xlab = 'i (iteration)',
|
matplot(loss.history, type = 'l', log = 'y', xlab = 'i (iteration)',
|
||||||
main = paste('loss', name),
|
main = paste('loss', name),
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue