Nothing
#' Wasserstein Distance by Entropic Regularization
#'
#' Due to high computational cost for linear programming approaches to compute
#' Wasserstein distance, \insertCite{cuturi_sinkhorn_2013;textual}{T4transport} proposed an entropic regularization
#' scheme as an efficient approximation to the original problem. This comes with
#' a regularization parameter \eqn{\lambda > 0} in the term
#' \deqn{\lambda h(\Gamma) = \lambda \sum_{m,n} \Gamma_{m,n} \log (\Gamma_{m,n}).}
#' As \eqn{\lambda\rightarrow 0},
#' the solution to an approximation problem approaches to the solution of a
#' true problem. However, we have an issue with numerical underflow. Our
#' implementation returns an error when it happens, so please use a larger number
#' when necessary.
#'
#' @param X an \eqn{(M\times P)} matrix of row observations.
#' @param Y an \eqn{(N\times P)} matrix of row observations.
#' @param D an \eqn{(M\times N)} distance matrix \eqn{d(x_m, y_n)} between two sets of observations.
#' @param p an exponent for the order of the distance (default: 2).
#' @param wx a length-\eqn{M} marginal density that sums to \eqn{1}. If \code{NULL} (default), uniform weight is set.
#' @param wy a length-\eqn{N} marginal density that sums to \eqn{1}. If \code{NULL} (default), uniform weight is set.
#' @param lambda a regularization parameter (default: 0.1).
#' @param ... extra parameters including \describe{
#' \item{maxiter}{maximum number of iterations (default: 496).}
#' \item{abstol}{stopping criterion for iterations (default: 1e-10).}
#' }
#'
#' @return a named list containing\describe{
#' \item{distance}{\eqn{\mathcal{W}_p} distance value.}
#' \item{iteration}{the number of iterations it took to converge.}
#' \item{plan}{an \eqn{(M\times N)} nonnegative matrix for the optimal transport plan.}
#' }
#'
#' @examples
#' \donttest{
#' #-------------------------------------------------------------------
#' # Wasserstein Distance between Samples from Two Bivariate Normal
#' #
#' # * class 1 : samples from Gaussian with mean=(-1, -1)
#' # * class 2 : samples from Gaussian with mean=(+1, +1)
#' #-------------------------------------------------------------------
#' ## SMALL EXAMPLE
#' set.seed(100)
#' m = 20
#' n = 10
#' X = matrix(rnorm(m*2, mean=-1),ncol=2) # m obs. for X
#' Y = matrix(rnorm(n*2, mean=+1),ncol=2) # n obs. for Y
#'
#' ## COMPARE WITH WASSERSTEIN
#' outw = wasserstein(X, Y)
#' skh1 = sinkhorn(X, Y, lambda=0.05)
#' skh2 = sinkhorn(X, Y, lambda=0.10)
#'
#' ## VISUALIZE : SHOW THE PLAN AND DISTANCE
#' pm1 = paste0("wasserstein plan ; distance=",round(outw$distance,2))
#' pm2 = paste0("sinkhorn lbd=0.05; distance=",round(skh1$distance,2))
#' pm5 = paste0("sinkhorn lbd=0.1 ; distance=",round(skh2$distance,2))
#'
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' image(outw$plan, axes=FALSE, main=pm1)
#' image(skh1$plan, axes=FALSE, main=pm2)
#' image(skh2$plan, axes=FALSE, main=pm5)
#' par(opar)
#' }
#'
#' @references
#' \insertAllCited{}
#'
#' @concept dist_wass
#' @name sinkhorn
#' @rdname sinkhorn
NULL
#' @rdname sinkhorn
#' @export
sinkhorn <- function(X, Y, p=2, wx=NULL, wy=NULL, lambda=0.1, ...){
## INPUTS : EXPLICIT
if (is.vector(X)){
X = matrix(X, ncol=1)
}
if (is.vector(Y)){
Y = matrix(Y, ncol=1)
}
if (!is.matrix(X)){ stop("* sinkhorn : input 'X' should be a matrix.") }
if (!is.matrix(Y)){ stop("* sinkhorn : input 'Y' should be a matrix.") }
if (base::ncol(X)!=base::ncol(Y)){
stop("* sinkhorn : input 'X' and 'Y' should be of same dimension.")
}
m = base::nrow(X)
n = base::nrow(Y)
wxname = paste0("'",deparse(substitute(wx)),"'")
wyname = paste0("'",deparse(substitute(wy)),"'")
fname = "sinkhorn"
par_wx = valid_weight(wx, m, wxname, fname)
par_wy = valid_weight(wy, n, wyname, fname)
par_p = max(1, as.double(p))
par_D = as.matrix(compute_pdist2(X, Y))
par_lbd = max(sqrt(.Machine$double.eps), as.double(lambda))
## INPUTS : IMPLICIT
params = list(...)
pnames = names(params)
if ("maxiter" %in% pnames){
par_iter = max(2, round(params$maxiter))
} else {
par_iter = 496
}
if ("abstol" %in% pnames){
par_tol = max(100*.Machine$double.eps, as.double(params$abstol))
} else {
par_tol = 1e-10
}
# ## MAIN COMPUTATION
output = cpp_sinkhorn13(par_wx, par_wy, par_D, par_lbd, par_p, par_iter, par_tol)
return(output)
}
#' @rdname sinkhorn
#' @export
sinkhornD <- function(D, p=2, wx=NULL, wy=NULL, lambda=0.1, ...){
## INPUTS : EXPLICIT
name.fun = "sinkhornD"
name.D = paste0("'",deparse(substitute(D)),"'")
name.wx = paste0("'",deparse(substitute(wx)),"'")
name.wy = paste0("'",deparse(substitute(wy)),"'")
par_D = valid_distance(D, name.D, name.fun)
par_wx = valid_weight(wx, base::nrow(D), name.wx, name.fun)
par_wy = valid_weight(wy, base::ncol(D), name.wy, name.fun)
par_p = max(1, as.double(p))
par_lbd = max(sqrt(.Machine$double.eps), as.double(lambda))
## INPUTS : IMPLICIT
params = list(...)
pnames = names(params)
par_iter = max(1, round(ifelse((("maxiter")%in%pnames), params$maxiter, 496)))
par_tol = max(sqrt(.Machine$double.eps), as.double(ifelse(("abstol"%in%pnames), params$abstol, 1e-10)))
# ## MAIN COMPUTATION
output = cpp_sinkhorn13(par_wx, par_wy, par_D, par_lbd, par_p, par_iter, par_tol)
return(output)
}
# m = sample(100:200, 1)
# n = sample(100:200, 1)
# X = matrix(rnorm(m*2, mean=-1),ncol=2) # m obs. for X
# Y = matrix(rnorm(n*2, mean=+1),ncol=2) # n obs. for Y
#
# ## COMPUTE WITH DIFFERENT ORDERS
#
# sqrt(8)
# wasserstein(X, Y)$distance
# sinkhorn(X, Y, lambda=0.001)$distance
# sinkhorn(X, Y, lambda=0.005)$distance
# sinkhorn(X, Y, lambda=0.01)$distance
# sinkhorn(X, Y, lambda=0.05)$distance
# sinkhorn(X, Y, lambda=0.1)$distance
# sinkhorn(X, Y, lambda=1)$distance
# sinkhorn(X, Y, lambda=10)$distance
# sinkhorn(X, Y, lambda=100)$distance
# sinkhorn(X, Y, lambda=1000)$distance
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.